The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

THE HALF-W A VE MAPS EQUA TION ON T : GLOBAL WELL-POSEDNESS IN H 1 / 2 AND ALMOST PERIODICITY P A TRICK G ´ ERARD AND ENNO LENZMANN Abstract. W e consider the half-wa ve maps equation ∂ t u = u × | D | u for u : R × T → S 2 , where T = R / 2 π Z is the one-dimensional torus and S 2 ⊂ R 3 denotes the unit sphere. By extension from rational initial data, w e construct a unique and contin uous ﬂow map for data in the critical energy space H 1 / 2 ( T ; S 2 ). Moreov er, w e show almost p eriodicity in time of these solutions. F or the dense subset of rational initial data, we establish quasi-p erio dicit y in time and a-priori bounds on ∥ u ( t ) ∥ H s ( T ) for any s > 0. Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of v ector-v alued holomorphic functions on the unit disk. As a central ingredient, we develop a general stability principle for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin–Ono equation, Calogero–Sutherland DNLS, and the half-wa ve-maps equation p osed on T . Our results extend to the matrix-v alued half-wa v e maps equation ∂ t U = − i 2 [ U , | D | U ] with target manifold given b y the complex Grassmannians Gr k ( C d ), thereby generalizing the sp ecial case S 2 ∼ = CP 1 ∼ = Gr 1 ( C 2 ). In a companion w ork, we prov e global well-posedness for the half-wa ve maps equation posed on R in the scaling-critical energy space ˙ H 1 / 2 , b y establishing a stabilit y principle for explicit formulae on Hardy spaces in the complex half-plane C + . Contents 1. In tro duction 2 2. Main results 5 3. Road map and remarks 8 4. Notation and preliminaries 12 5. Sp ectral analysis of T U 16 6. Lax pair structure 19 7. Global well-posedness for rational data 23 8. Stabilit y principle for explicit form ulae 27 9. Global well-posedness in H 1 / 2 32 10. Almost Periodicity 37 App endix A. L WP in H s with s > 3 2 41 App endix B. Half-harmonic maps and solitary wa ves 42 References 43 1 2 P A TRICK G ´ ERARD AND ENNO LENZMANN 1. In tro duction 1.1. F orm ulation of the problem. As a starting point of this pap er, we consider the half-w av e maps equation (HWM) p osed on the one-dimensional torus T = R / 2 π Z with target S 2 . The corresp onding evolution equation can b e written as ∂ t u = u × | D | u (HWM S 2 ) for the map u : R × T → S 2 . Here S 2 is the standard unit t wo-sphere em b edded in R 3 and × stands for the vector pro duct in R 3 . As usual, the op erator | D | denotes the ﬁrst-order fractional deriv ative for functions on T , i.e., | D | f ( θ ) = ∞ X n = −∞ | n | b f n e inθ for θ ∈ T . W e refer to [ 27, 17, 19, 3, 18, 13 ] for recent works on (HWM) and its related v ersion posed on R . The half-wa v e maps equation was introduced indep endently in [ 17, 27 ], mo- tiv ated by the theory of half-harmonic maps and b y connections with integrable Calogero–Moser spin systems. It is a Hamiltonian equation whose energy func- tional is E [ u ] = 1 2 Z T u · | D | u = 1 2 ∞ X n = −∞ | n || b u n | 2 = ∥ u ∥ 2 ˙ H 1 / 2 . The natural energy space is therefore H 1 / 2 ( T ; S 2 ) = { u ∈ H 1 / 2 ( T ; R 3 ) : u ( θ ) ∈ S 2 a.e. } . Geometrically , this is a lo op space, and (HWM) may be view ed as a Hamiltonian ﬂo w on the inﬁnite-dimensional symplectic space H 1 / 2 -lo ops in to S 2 . The energy is c onformal ly invariant . Indeed, writing E [ u ] = 1 4 π Z Z T × T | u ( θ ) − u ( φ ) | 2 2 sin 2 ( θ − φ 2 ) dθ dφ = 1 4 π Z D |∇ u e ( x, y ) | 2 dx dy, where u e : D → R 3 denotes the harmonic extension of u ∈ H 1 / 2 ( R ; S 2 ) to the unit disk D , one obtains E [ u ◦ φ ] = E [ u ] for every conformal automorphism φ : D → D . Critical points of E are referred to as half-harmonic maps , whose regularit y theory w as dev elop ed in the seminal w ork of Rivi` ere–da Lio [ 6 ]; see also [ 21, 17, 20 ]. Lo cal well-posedness of (HWM) for H s -data with s > 3 2 follo ws by an itera- tiv e sc heme for hyperb olic systems. How ever, despite the complete integrabilit y of (HWM), the global w ell-p osedness of the Cauch y problem has remained op en, even for smo oth data suﬃciently close to a constant state. The main obstacles are: • The equation is a quasi-line ar system. • The op erator | D | pro vides no dispersion in one space dimension. • The Lax structure for (HWM) do es not control Sob olev norms ab o ve H 1 / 2 . One of the principal con tributions of this w ork is to bridge this gap by pro ving global well-posedness for (HWM) in the critical energy space H 1 / 2 . The construc- tion pro ceeds by ﬁrst proving global well-posedness for all rational initial data, where we adapt our recen t approach in [ 13 ] from R to T . Now, the densit y of ratio- nal data in the energy space H 1 / 2 com bined with an explicit form ula for (HWM) HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 3 enables us to obtain a unique extension yielding weak solutions that could exhibit loss of energy . Ho wev er, to construct a decent ﬂow map for H 1 / 2 -data, we need to pro ve its strong con tinuit y , which is equiv alent to proving energy conserv ation. Here, a key ingredient will b e a newly found stability principle for explicit for- m ulae asso ciated with completely integrable PDEs on Hardy spaces, suc h as the Benjamin–Ono equation (BO), the Calogero–Sutherland DNLS (CS-DNLS), the cubic Szeg˝ o and (HWM) in [ 10, 11, 13, 14, 1, 15, 16, 9, 5, 22 ]. 1.2. Matrix-v alued generalization of (HWM). Before we state our main re- sults, it will b e useful to introduce the following generalization of the half-wa ve maps equation beyond the target S 2 . By follo wing [ 12 ], w e notice that (HWM S 2 ) can b e rephrased in matrix-commutator form giv en b y ∂ t U = − i 2 [ U , | D | U ] (HWM) with the matrix-v alued map U = u · σ = 3 X k =1 u k σ k =  u 3 u 1 − iu 2 u 1 + iu 2 − u 3  , where σ = ( σ 1 , σ 2 , σ 3 ) denote the standard Pauli matrices. It is straigh tforward (see [ 12 ]) to verify that the geometric constrain t that u ( t, · ) tak es v alues in S 2 is equiv alen t to the algebraic conditions U ( t, · ) = U ( t, · ) ∗ , U ( t, · ) 2 = 1 2 , T r( U ( t, · )) = 0 . As recen tly proposed in [ 13 ], the matrix-v alued form ulation of (HWM) leads to a natural generalization by considering maps U v alued in the c omplex Gr assmanni- ans , which w e denote by Gr k ( C d ) and identify with the set of matrices Gr k ( C d ) :=  U ∈ C d × d : U ∗ = U, U 2 = 1 d , T r( U ) = d − 2 k  , where d ≥ 2 and 0 ≤ k ≤ d are given integers. 1 F or any U ∈ Gr k ( C d ), w e note that P = 1 2 ( 1 d − U ) is an orthogonal pro jection P = P ∗ = P 2 with Rank( P ) = T r( P ) = k . Th us, we can canonically iden tify elemen ts in Gr k ( C d ) with the k -dimensional subspaces in C d in accordance with the geometric deﬁnition of the Grassmannian Gr k ( C d ). F urthermore, we recall that Gr k ( C d ) is a compact K¨ ahler manifold of complex dimension k ( d − k ) and the Grassmannians Gr 1 ( C d ) clearly correspond to the pro jectiv e spaces CP d − 1 whic h are particularly interesting from the physical p oin t of view. F or the rest of this pap er, we will th us study (HWM) with the targets Gr k ( C d ), where w e remind the reader that all results shown below also apply to (HWM S 2 ) b y using that U = u · σ ∈ Gr 1 ( C 2 ) ∼ = CP 1 with u = ( u 1 , u 2 , u 3 ) ∈ S 2 if and only if u k = 1 2 T r( U σ k ) for k = 1 , 2 , 3. 1.3. Lax pair structure. F or (HWM) with target Gr k ( C d ), the energy functional is readily found to be E [ U ] = 1 2 Z T T r( U ( | D | U )) dθ = 1 2 ∞ X n = −∞ | n || b U n | 2 = 1 2 ∥ U ∥ 2 ˙ H 1 / 2 (1.1) 1 Note the trivial cases Gr 0 ( C d ) = { 1 d } and Gr d ( C d ) = {− 1 d } . T o streamline the presentation, we will mostly consider the non-trivial cases Gr k ( C d ) with 1 ≤ k ≤ d − 1. 4 P A TRICK G ´ ERARD AND ENNO LENZMANN with the corresp onding energy space H 1 / 2 ( T ; Gr k ( C d )) = { U ∈ H 1 / 2 ( T ; C d × d ) : U ( θ ) ∈ Gr k ( C d ) for a.e. θ ∈ T } . As before, we notice that the energy E [ U ] is conformally inv arian t and hence the energy space H 1 / 2 ( T ; Gr k ( C d )) is critical for (HWM) with resp ect to this inv ariance of the problem. F rom the analysis of half-harmonic maps, we infer that (HWM) has non-trivial stationary solutions and more generally tr aveling solitary waves , see the app endix for details. Let us now brieﬂy discuss the Lax pair structure b ehind the half-w av e maps equation. T o this end, w e assume that U ∈ C ( I ; H s ) for some s > 3 2 solv es (HWM) on some time in terv al I with 0 ∈ I . F or any t ∈ I , we consider the self-adjoint T o eplitz op erator T U ( t ) : L 2 + → L 2 + , F 7→ T U ( t ) F = Π( U ( t ) F ) (1.2) deﬁned in the matrix-v alued Hardy space L 2 + = L 2 + ( T ; C d × d ), where Π denotes the Cauc hy–Szeg˝ o pro jection onto L 2 + . By follo wing [ 12, 13 ], w e exploit the algebraic iden tities for the matrix-v alued map U together with p ro duct iden tities for T oeplitz op erators to conclude that the following L ax e quation holds: d dt T U ( t ) = [ B U ( t ) , T U ( t ) ] for t ∈ I . (1.3) Here the unbounded op erator B U ( t ) = − i 2 ( T U ( t ) D + D T U ( t ) ) + i 2 T | D | U ( t ) is essen tially skew-adjoin t with Dom( B U ( t ) ) = { F ∈ L 2 + : D F ∈ L 2 + } and D = − i∂ θ . As a tec hnical aside, w e remark that B U ( t ) is neither b ounded from below nor abov e, i.e., its sp ectral prop erties resemble that of a Dirac-type op erator and, moreo ver, its (essen tial) sk ew-adjointness cannot b e inferred from the Kato–Rellic h theorem due to ∥ T U ∥ = 1. W e refer to Appendix A for details. By (1.3), w e infer the Lax evolution of T U ( t ) b y the unitary equiv alence T U ( t ) = U ( t ) T U 0 U ( t ) ∗ (1.4) with U ( t ) being the unitary map that solves the op erator diﬀerenti al equation d dt U ( t ) = B U ( t ) U ( t ) for t ∈ I and U (0) = I . Eviden tly , the sp ectrum of T U ( t ) is preserved for t ∈ I . T o extract a useful set of conserv ed quan tities, w e observ e that the algebraic prop erty U ( t, · ) 2 = 1 d implies T 2 U ( t ) = I − K U ( t ) with K U ( t ) = H ∗ U ( t ) H U ( t ) , with the Hankel op erator H U ( t ) F = Π − ( U ( t ) F ) for F ∈ L 2 + with the pro jection Π − = I − Π onto L 2 − = L 2 ⊖ L 2 + . Now, a direct calculation in Section 5 below yields E [ U ( t )] = 1 2 ∥ U ( t ) ∥ 2 ˙ H 1 / 2 = T r( K U ( t ) ) . Th us the op erator K U ( t ) is trace-class if and only if U ( t ) ∈ H 1 / 2 , and from (1.4) w e obtain the unitary equiv alence K U ( t ) = U ( t ) K U 0 U ( t ) ∗ , (1.5) HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 5 whic h reprov es conserv ation of energy E [ U ( t )] = E [ U 0 ] for t ∈ I , pro vided that U ( t ) is a suﬃciently smo oth solution. In fact, from (1.5) we can infer that suﬃ- cien tly smo oth solutions of (HWM) exhibit the following family of conserved quan- tities given b y I p [ U ] = T r( | K U | p/ 2 ) = T r( | H U | p ) for all p ∈ [1 , ∞ ) corresp onding to the Schatten- p -norms ∥ H U ∥ S p = T r( | H U | p ) 1 /p . By inv oking P eller’s theorem (see e.g. [ 24 ]), we obtain the a-priori b ounds sup t ∈ I ∥ U ( t ) ∥ B 1 /p p,p ≲ ∥ U (0) ∥ B 1 /p p,p for all p ∈ [1 , ∞ ) (1.6) in the class of Besov spaces B 1 /p p,p ( T ; C d × d ), where we recall that B 1 / 2 2 , 2 = H 1 / 2 . Ho wev er, these bounds are still oﬀ b y more than one deriv ativ e needed to deduce global well-posedness for solutions in H s with s > 3 2 . Th us, at the moment, it is far from clear how to obtain a lo cal w ell-p osedness theory for (HWM) such that the a-priori b ounds (1.6) can b e used to obtain global regularit y . 2 Finally , we mention the remark able feature that the half-w av e maps equation preserv es r ationality of solutions . That is, if the initial datum U 0 : T → Gr k ( C d ) is a rational function of z = e iθ with θ ∈ T , then the corresp onding smo oth solution U ∈ C ( I ; H ∞ ) is a rational function U ( t ) : T → Gr k ( C d ) for any t ∈ I on its maximal time in terv al I . Indeed, this feature follows from the unitary equiv alence (1.5) by the Lax ev olution together with the fact that K U ( t ) has ﬁnite rank if and only if U ( t ) : T → Gr k ( C d ) is rational , whic h is a direct consequence of Kr one cker’s the or em for the Hank el operator H U ( t ) . In fact, the preserv ation of rationalit y will b e our starting p oint for pro ving global- in-time existence of rational solutions of (HWM), whic h later will b e used to con- struct the aforementioned ﬂo w map with data in H 1 / 2 . Ac knowledgmen ts. P . G´ erard w as partially supp orted b y the F rench Agence Na- tionale de la Recherc he under the ANR pro ject ISAAC–ANR-23–CE40-0015-01. E. Lenzmann gratefully ackno wledges ﬁnancial supp ort from Swiss National Sci- ence F oundation (SNSF) through Gran t No. 204121. 2. Main results In this section, w e will state our main results that sho w global well-posedness of (HWM) in the critical energy space H 1 / 2 ( T ; Gr k ( C d )) and their almost perio dicit y in time. 2.1. Global w ell-p osedness in H 1 / 2 . As a starting p oin t, we observe that an adaptation of the strategy in [ 13 ] yields the follo wing global w ell-p osedness result for all rational initial data, whic h we denote b y R at ( T ; Gr k ( C d )) henceforth. Lemma 2.1 (GWP for rational data) . L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers. Then, for every U 0 ∈ R at ( T ; Gr k ( C d )) , ther e exists a unique solution U ∈ C ∞ ( R × T ) of (HWM) with initial datum U (0) = U 0 . 2 A slighlt y reﬁned analysis sho ws that an a-priori bound on ∥ ∂ θ U ( t ) ∥ L ∞ would b e suﬃcien t to conclude global existence. Ho wev er, b y Peller’s result and the Lax structure, we can only bound ∥ ∂ θ U ( t ) ∥ L 1 ≲ ∥ U (0) ∥ B 1 1 , 1 . 6 P A TRICK G ´ ERARD AND ENNO LENZMANN Mor e over, we have c onservation of the me an and the ener gy, i.e., M [ U ( t )] = M [ U 0 ] = 1 2 π Z T U 0 , E [ U ( t )] = E [ U 0 ] for al l t ∈ R , and it holds that U ( t ) ∈ R at ( T ; Gr k ( C d )) for al l t ∈ R . Next, w e recall that R at ( T ; Gr k ( C d )) is dense in H 1 / 2 ( T ; Gr k ( C d )), see Theorem 4.1 b elo w from [ 13 ]. Th us by taking limits of rational solutions, we easily construct global weak solutions U ∈ C ( R ; H 1 / 2 w ) with E [ U ( t )] ≤ E [ U 0 ] for all t ∈ R of (HWM) for initial data U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )), where H 1 / 2 w denotes H 1 / 2 equipp ed with its weak topology . Our ﬁrst main result pro vides a substan tial strengthening b y sho wing uniqueness of this weak limit and energy conserv ation, i.e., we ha ve the equalit y E [ U ( t )] = E [ U 0 ] for all t ∈ R , whic h yields the strong con tin uity of U ∈ C ( R ; H 1 / 2 ). W e formulate this as follo ws. Theorem 2.1 (GWP in H 1 / 2 ) . L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers. Then (HWM) with tar get Gr k ( C d ) is glob al ly wel l-p ose d in H 1 / 2 in the fol lowing sense. Ther e exists a unique c ontinuous map Φ : R × H 1 / 2 ( T ; Gr k ( C d )) → H 1 / 2 ( T ; Gr k ( C d )) , ( t, U 0 ) 7→ Φ t ( U 0 ) such that, for every U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) , the fol lowing pr op erties hold. (i) Solution: The map t 7→ Φ t ( U 0 ) b elongs to C ( R ; H 1 / 2 ( T ; Gr k ( C d ))) and solves (HWM) in the we ak sense with initial datum Φ t =0 ( U 0 ) = U 0 . (ii) Me an and ener gy c onservation: F or al l t ∈ R , it holds that M [Φ t ( U 0 )] = M [ U 0 ] , E [Φ t ( U 0 )] = E [ U 0 ] . (iii) Continuous dep endenc e: If U 0 ,n ∈ H 1 / 2 ( T ; Gr k ( C d )) is a se quenc e w ith U 0 ,n → U 0 in H 1 / 2 as n → ∞ , then Φ t ( U 0 ,n ) → Φ t ( U 0 ) in H 1 / 2 as n → ∞ lo c al ly uniformly in time. (iv) Gr oup pr op erty: It holds th at Φ t + s ( U 0 ) = Φ t (Φ s ( U 0 )) for al l t, s ∈ R . (v) Pr eservation of r ationality: F or r ational initial data U 0 ∈ R at ( T ; Gr k ( C d )) , we have Φ t ( U 0 ) ∈ R at ( T ; Gr k ( C d )) for al l t ∈ R . Based on the previous theorem, w e will refer to the data-to-solution map Φ ab o ve as the ﬂow map for (HWM) with initial data in H 1 / 2 . The next result sho ws that the corresp onding global-in-time solutions U ( t ) = Φ t ( U 0 ) from ab o ve still satisfy a Lax ev olution, initially found for smo oth solutions on short time in terv als. Note that, due to the lo w regularit y of solutions, the Lax equation (1.3) in comm utator form becomes meaningless as it is unclear ho w to deﬁne B U ( t ) if we only assume U ( t ) ∈ H 1 / 2 . Ho wev er, w e can still deduce that the T o eplitz operator T U ( t ) ev olves by a unitary equiv alence in the sense of a Lax ev olution. HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 7 Theorem 2.2 (Lax evolution) . Denote U ( t ) = Φ t ( U 0 ) for U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) with Φ as in The or em 2.1 ab ove. Then, for al l t ∈ R , ther e exists a unitary map U ( t ) : L 2 + ( T ; C d × d ) → L 2 + ( T ; C d × d ) such that T U ( t ) = U ( t ) T U 0 U ( t ) ∗ with the T o eplitz op er ator T U ( t ) : L 2 + ( T ; C d × d ) → L 2 + ( T ; C d × d ) deﬁne d in (1.2) . R emarks. 1) W e will prov e Theorem 2.2 b efore completing the pro of of Theorem 2.1 b elo w, as it provides us with a conv enien t wa y for proving energy conserv ation via the identit y E [ U ( t )] = T r( K U ( t ) ) together with the unitary equiv alence for K U ( t ) = U ( t ) K U 0 U ( t ) ∗ for the trace-class op erator K U ( t ) = I − T 2 U ( t ) . 2) The unitary map U ( t ) is giv en b y a so-called explicit formula whic h w e will discuss in Section 3 b elo w. Also, the explicit form ula for U ( t ) will pla y a k ey role in constructing the ﬂow map Φ in Theorem 2.1 abov e. 2.2. Long-time b eha vior of solutions. Our second main result shows that the solutions for (HWM) obtained by Theorem 2.1 are almost p erio dic in time; see Section 10 for a brief recap of almost p eriodic functions v alued in Banac h spaces. Theorem 2.3 (Almost p erio dicit y) . L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers. F or every U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) , the map t 7→ Φ t ( U 0 ) ∈ C ( R ; H 1 / 2 ) is almost p erio dic in time. As a c onse quenc e, we have the fol lowing pr op erties. (i) Poinc ar ´ e r e curr enc e holds in the sense that, for al l ε, T > 0 , ther e exists some time t ∗ ≥ T such that ∥ Φ t ∗ ( U 0 ) − U 0 ∥ H 1 / 2 ≤ ε . (ii) The orbit { Φ t ( U 0 ) : t ∈ R } is r elatively c omp act in H 1 / 2 . In the case of rational initial data, we can actually obtain the follo wing reﬁnemen t b y sho wing quasi-perio dicity in time, whic h corresp onds to a linear ﬂow on some torus T N = ( R / 2 π Z ) N . In particular, we obtain a-priori bounds on all Sob olev norms for rational solutions of (HWM). Here is the precise statemen t. Theorem 2.4 (Quasi-p erio dicit y and a-priori bounds for rational data) . L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers. F or every r ational initial datum U 0 ∈ R at ( T ; Gr k ( C d )) , the map t 7→ Φ t ( U 0 ) is quasi-p erio dic in time. That is, it holds that Φ t ( U 0 ) = G ( t ω ) for al l t ∈ R , wher e G : T N → R at ( T ; Gr k ( C d )) is some map dep ending only on U 0 with some inte ger N ≥ 1 and some c onstant ω ∈ R N . In p articular, we obtain the a-priori b ounds sup t ∈ R ∥ Φ t ( U 0 ) ∥ H s ≲ U 0 ,s 1 for al l s ≥ 0 . R emark. Note that the a-priori b ounds on all H s -norms for rational solutions of (HWM) obtained abov e are a consequence of their quasi-perio dicity in time – and not deduced from a family of conserved quan tities. A natural op en question is to gain insigh t into the b eha vior of H s -norms with s > 1 2 for non-rational initial data. More ambitiously , it w ould be interesting to pro v e (or dispro ve) global well- p osedness of (HWM) in H s for s > 1 2 . 8 P A TRICK G ´ ERARD AND ENNO LENZMANN 3. Road map and remarks 3.1. GWP for rational data. As the initial step tow ards proving Theorem 2.1, we consider rational initial data U 0 ∈ R at ( T ; Gr k ( C d )). Using the lo cal well-posedness in H s for any s > 3 2 , w e let U ∈ C ( I ; H s ) denote the corresp onding solution of (HWM) with U (0) = U 0 deﬁned on its maximal time in terv al of existence I . Now, the analysis of the Lax pair structure on the Hardy space L 2 + = L 2 + ( T ; C d × d ) allo ws to deduce that U can be expressed in terms of the explicit formula giv en by Π U ( t, z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  for ( t, z ) ∈ I × D (EF) where T U 0 denotes the T oeplitz op erator deﬁned in (1.2). Here we recall that Π denotes the Cauc hy–Szeg˝ o pro jection onto L 2 + and the operator M is the mean, whereas S ∗ stands for the bac kward (or righ t) shift operator on L 2 + . T o further clarify the meaning of (EF), we remark that w e use the canonical identiﬁcation of elements F ∈ L 2 + ( T ; C d × d ) with holomorphic functions F : D → C d × d with F ( z ) = P n ≥ 0 b F n z n suc h that P n ≥ 0 | b F n | 2 E < ∞ , where | · | E denotes the F robenius norm of matrices in C d × d ; see Section 4 below for a recap on vector-v alued Hardy spaces. The in terested reader will also ﬁnd b elow some discussion on the relation of (EF) to other explicit form ulae that ha ve b een found for completely integrable PDEs, which w as initiated by the work of the ﬁrst author of the present pap er. T o show global existence, i.e., w e hav e that I = R holds, it suﬃces to obtain an a-priori b ound sup t ∈ I ∥ U ( t ) ∥ H s < ∞ (3.1) for an y giv en s > 3 2 . T o ac hieve this goal, w e can adapt the strategy in [ 13 ] for the half-w av e maps equation R for rational initial data. More precisely , it turns out that the rationalit y of U 0 ∈ R at ( T ; Gr k ( C d )) in concert with Kronec k er’s theorem for Hankel op erators allows us to single out a ﬁnite-dimensional subspace K ⊂ L 2 + with the following properties. 1) Π U 0 ∈ K . 2) ( I − z e − itT U 0 S ∗ ) − 1 : K → K for all t ∈ R and z ∈ D . Hence we see that the analysis of (EF) for rational data completely reduces to matters in some ﬁnite-dimensional subspace (and the corresp onding dynamics could b e describ ed by a ﬁnite system of nonlinear ODEs). In particular, the question of obtaining a-priori b ounds b oils do wn to ruling out p oten tial eigen v alues of the con traction e − itT U 0 S ∗ : K → K on the unit circle for all times, i.e., σ p (e − itT U 0 S ∗ | K ) ∩ ∂ D = ∅ for all t ∈ R . (3.2) Indeed, we can derive (3.2) b y exploiting comm utator prop erties of T U 0 with the bac kward shift S ∗ . Once the sp ectral prop ert y (3.2) has been established, w e obtain strong uniform b ounds on the resolven t ( I − z e − itT U 0 S ∗ ) − 1 on K . By going bac k to (EF), this allows us to deduce the desired a-priori b ound (3.1) and we th us obtain global existence for all rational data. In fact, a slightly more reﬁned analysis of (EF) yields the quasi-p erio dicit y for rational solutions in Theorem 2.4 ab ov e. 3.2. GWP in H 1 / 2 . F or non-rational data U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )), we choose a sequence U 0 ,n ∈ R at ( T ; Gr k ( C d )) such that U 0 ,n → U 0 in H 1 / 2 thanks to the densit y result in Theorem 4.1. Let U n ∈ C ∞ ( R × T ) denote the corresponding global smo oth solutions of (HWM) with initial datum U n (0) = U 0 ,n . Next, b y in voking HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 9 the explicit formula (EF) v alid for eac h U n ( t ) and by using energy conserv ation for these smo oth solutions, we can deduce that U n ( t ) → U ( t ) in L 2 and U n ( t ) ⇀ U ( t ) in H 1 / 2 for any t ∈ R . (3.3) Here the limit U ∈ C ( R ; H 1 / 2 w ) is a weak solution of (HWM) with U (0) = U 0 and it is given b y the explicit form ula Π U ( t, z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  for ( t, z ) ∈ R × D , (EF) whic h in particular shows that the limit U ( t ) is indep enden t of the chosen sequence of appro ximating rational data U 0 ,n . Ho wev er, the w eak con vergence in H 1 / 2 only guaran tees the inequalit y E [ U ( t )] ≤ E [ U 0 ] for all t ∈ R . No w, the ma jor key step tow ards proving global well-posedness in the sense of Theorem 2.1 rests on sho wing that energy conserv ation holds, i.e., w e hav e E [ U ( t )] = E [ U 0 ] for all t ∈ R and thus the limit U ∈ C ( R ; H 1 / 2 ) is in fact con tin uous. Ho wev er, the analysis of (EF) for non-rational data U 0 ∈ H 1 / 2 is substantially more complicated, since the analogue of the subspace K ⊂ L 2 + from before must b e inﬁnite-dimensional due to Kroneck er’s theorem. In particular, the sp ectral prop ert y (3.2) – although also true for non-rational U 0 – is far from suﬃcien t to get enough con trol from (EF) directly . T o ov ercome this obstacle, we devise the follo wing s trategy . Giv en a sequence U 0 ,n ∈ R at ( T ; Gr k ( C d )) with U 0 ,n → U 0 in H 1 / 2 , we in tro duce the following maps U n ( t ) : L 2 + → L 2 + b y setting ( U n ( t ) F )( z ) := M  ( I − z e − itT U 0 ,n S ∗ ) − 1 F  (3.4) for t ∈ R , z ∈ D , and F ∈ L 2 + . Likewise, w e deﬁne the map U ( t ) : L 2 + → L 2 + as ( U ( t ) F )( z ) := M  ( I − z e − itT U 0 S ∗ ) − 1 F  . (3.5) By the Lax evolution for the smooth rational solutions U n ∈ C ∞ ( R × T ), w e can deduce the following properties for all t ∈ R and n ∈ N : 1) ∂ t U n ( t ) = B U n ( t ) U n ( t ) with U n (0) = I . 2) U n ( t ) : L 2 + → L 2 + is unitary . 3) T U n ( t ) U n ( t ) = U n ( t ) T U 0 ,n . Note that the con vergence properties in (3.3) imply that T U n ( t ) → T U ( t ) strongly as op erators for any t ∈ R , whic h is seen to yield, for ev ery t ∈ R , that U n ( t ) ⇀ U ( t ) weakly as op erators . Moreo ver, by using the strong op erator conv ergence of the self-adjoin t operators T U n ( t ) , we can pass to the limit in (3) to deduce the intert wining relation T U ( t ) U ( t ) = U ( t ) T U 0 for all t ∈ R . (3.6) No w the central question is whether the map U ( t ) : L 2 + → L 2 + is unitary for all t ∈ R ? If so, then we can use (3.6) to infer the unitary equiv alence K U ( t ) = U ( t ) T U 0 U ( t ) ∗ for all t ∈ R 10 P A TRICK G ´ ERARD AND ENNO LENZMANN for the trace-class op erator K U ( t ) = I − T 2 U ( t ) . Once this is established, w e can directly deduce the desired energy conserv ation in view of E [ U ( t )] = T r( K U ( t ) ) = T r( K U 0 ) = E [ U 0 ]. T o establish that the limiting map U ( t ) is indeed unitary , we deriv e the follo wing stability principle for explicit formulae in a general setting that go es b eyond the sp eciﬁcs of (HWM). Theorem (Stability principle for explicit form ulae) . L et E b e a c omplex Hilb ert sp ac e and supp ose that L : Dom( L ) ⊂ L 2 + ( T ; E ) → L 2 + ( T ; E ) is a (p ossibly un- b ounde d) self-adjoint op er ator. F or t ∈ R , we deﬁne the map U L ( t ) : L 2 + ( T ; E ) → L 2 + ( T ; E ) by setting ( U L ( t ) F )( z ) := M  ( I − z e − itL S ∗ ) − 1 F  with z ∈ D . Then U L ( t ) is unitary for al l t ∈ R if and only if one of the fol lowing e quivalent c onditions holds. (i) Ker U L ( t ) = { 0 } for al l t ∈ R . (ii) lim j →∞ ∥ (e − itL S ∗ ) j F ∥ L 2 = 0 for al l t ∈ R and F ∈ L 2 + ( T ; E ) . R emarks. 1) The pro of of this theorem rests on some operator-theoretic analysis, whic h connects to the circle of ideas of the W old decomposition for isometries on Hilb ert spaces. How ever, we will as sume no op erator-theoretic background and pro vide a self-con tained approach. 2) Condition (ii) means that the discrete semigroup { (e − itL S ∗ ) j } j ∈ N is str ongly stable for each t ∈ R . This is a natural condition from an abstract op erator p oin t of view. Ow ed to this fact, we hav e c hosen to refer to the result ab o ve as the stability principle for explicit formulae. 3) The trivialit y of the k ernel of U L ( t ) stated in (i) can b e seen as a sort of non- de gener acy condition. In the con text of explicit formulae for completely integrable PDEs, this nondegeneracy seems easier to b e v eriﬁed than (ii). See in particular the pro of b elo w, where w e establish that Ker U ( t ) = { 0 } in the context of (HWM). 4) How ever, we remark that we can also ﬁnd examples where Ker U ( t )  = { 0 } o ccurs for some t ∈ R with a suitable choice for the self-adjoint op erator L . F or instance, this o ccurs for the explicit formula for the zer o-disp ersion limit for the Benjamin–Ono (BO) equation, whic h corresp onds to breakdown (sho ck formation) of solutions in ﬁnite time. In Subsection 3.4 b elo w, w e compare the explicit form ulae for the half-wa ve maps equation and the zero-disp ersion limit of (BO). 5) In a companion work on global well-posedness for (HWM) on the real line, w e deriv e an analogous stabilit y principle for explicit form ulae in the Hardy space L 2 + ( R ; E ). In view of the general result ab ov e, it remains to show that U ( t ) deﬁned in (3.5) satisﬁes Ker U ( t ) = { 0 } for all t ∈ R . Let us brieﬂy sketc h the pro of as follo ws. F rom the in tertwining relation (3.6) we see that eigen vectors F ∈ L 2 + of T U 0 are transp orted by U ( t ): If T U 0 F = µ F for some eigenv alue µ , then F ( t ) := U ( t ) F satisﬁes the equation T U ( t ) F ( t ) = µ F ( t ) . Ho wev er, the crucial step is now to show that F ( t ) is actually an eigenv ector of T U ( t ) as well, i.e., we ha ve F ( t )  = 0. Here, the sp eciﬁcs of the Lax op erator T U 0 HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 11 for (HWM) enter, where we mak e use of the crucial prop ert y that eigenspaces of T o eplitz op erators are ne arly S ∗ -invariant 3 . W e refer to Section 9 for details. Finally , once w e hav e shown that U ( t ) maps eigenfunctions of T U 0 to eigenfunc- tions of T U ( t ) , the triviality of Ker U ( t ) = { 0 } can b e derived by using that the sp ectrum of T U 0 is pure point, i.e., there is a basis of L 2 + consisting of eigenfunctions of T U 0 together with the fact that T U ( t ) is self-adjoint. 3.3. Almost perio dicity. W e now discuss how to pro ve almost p erio dicit y of the solution U ( t ) = Φ t ( U 0 ) ∈ C ( R ; H 1 / 2 ) with initial datum U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )). By Bo c hner’s criterion for almost p erio dicit y , it suﬃces to show that the set of translates of U ∈ C ( R ; H 1 / 2 ) deﬁned as T rans( U ) := { U ( · + a ) : a ∈ R } is relatively compact in the Banach space of b ounded and contin uous functions BC( R ; H 1 / 2 ) equipp ed with the sup-norm. T o sho w this prop erty , we make use of the explicit formula (EF) as follows. First, b y the construction of the ﬂow map Φ, w e hav e U ( t ) = Φ t ( U 0 ) = ( U ( t )Π U 0 ) + ( U ( t )Π U 0 ) ∗ − M ( U 0 ) . F or the sak e of the pro of, let us write the unitary map U ( t ) : L 2 + → L 2 + as ( U ( t ) F )( z ) = M  ( I − z Ω( t ) S ∗ ) − 1 F  for ( t, z ) ∈ R × D with the strongly contin uous unitary one-parameter group Ω( t ) := e − itT U 0 : L 2 + → L 2 + for t ∈ R . Since T U 0 has at most countable pure p oint sp ectrum, a standard Cantor diago- nalization argumen t yields the following c omp actness pr op erty : F or ev ery sequence ( a n ) in R , there exists a subsequence denoted by ( a ′ n ) such that Ω( a ′ n ) → Ω ∞ strongly as op erators with some unitary map Ω ∞ in L 2 + . Next, b y revisiting the analysis of the explicit form ula in the global well-posedness proof combined with the group prop ert y of Ω( t ), we can sho w that sup t ∈ R ∥ U ( t + a ′ n ) − U ∞ ( t ) ∥ H 1 / 2 → 0 as n → ∞ , where U ∞ ∈ C ( R ; H 1 / 2 ) is given b y the explicit-type form ula Π U ∞ ( t, z ) = M  ( I − z Ω( t )Ω ∞ S ∗ ) − 1 Π U 0  for ( t, z ) ∈ R × D . This chain of steps pro ves the relative compactness of T rans( U ) in BC( R ; H 1 / 2 ). As a ﬁnal remark, w e note that pro ving almost perio dicit y b y the explicit form ula only in volv es that the corresp onding Lax op erator L generating Ω( t ) = e − itL has at most countable pure p oint spectrum. Thus we exp ect our argumen ts to carry o ver to (BO) and (CS-DNLS) on T . 3 The nearly S ∗ -inv ariance prop erty can b e seen as a substitute for the lac k of elliptic regularit y for eigenfunctions of the zero-order operator T U 0 . By con trast, elliptic regularit y for eigenfunctions of the ﬁrst-order Lax op erators L for (BO) and (CS-DNLS) on T can b e used to gain control of Sobolev norms in the transp ort by U ( t ) to rule out the v anishing of F ( t ) = U ( t ) F ; see [ 1 ]. 12 P A TRICK G ´ ERARD AND ENNO LENZMANN 3.4. On the zero-disp ersion limit of (BO). W e remark that (EF) has a striking resem blance to the explicit form ula for the zero-disp ersion limit for (BO), ﬁrst deriv ed in [ 7 ] (and adapted in [ 22 ]) in the perio dic setting, and in [ 11 ] by the ﬁrst author of this paper on the line. More precisely , we recall from [ 7, 22 ] that for initial data u 0 ∈ L ∞ ( T ; R ) the zero-dispersion limit of (BO) denoted b y Z D [ u 0 ]( t ) ∈ C ( R ; L 2 w ( T ; R )) is giv en by the explicit formula (adapted to our notation) Π( Z D [ u 0 ])( t, z ) = M  ( I − z e − itT u 0 S ∗ ) − 1 Π u 0  with ( t, z ) ∈ R × D , where M = ⟨·| 1 ⟩ with the constan t function 1 corresponds to the mean in L 2 + ( T ; C ). Here T u 0 : L 2 + ( T ; C ) → L 2 + ( T ; C ) is the T oeplitz op erator with real-v alued sym b ol u 0 ∈ L ∞ ( T ; R ). Ho wev er, it is known that Z D [ u 0 ] can exhibit loss of L 2 -norm after the ﬁrst breaking time T + = − 1 2 min θ ∈ T u ′ 0 ( θ ) > 0 for an y non-constant initial datum u 0 ∈ C 1 ( T ; R ), which corresp onds to the onset of a sho ck at T + > 0 for the inviscid Burger’s equation ∂ t u + 2 u∂ θ u = 0 with u | t =0 = u 0 . Indeed, for u 0 ( θ ) = − cos θ and hence T + = 1 2 , w e deduce from [ 22 ] together with [ 8 ][Lemma 20] the strict inequalit y ∥ Π( Z D [ u 0 ])( t ) ∥ L 2 < ∥ Π u 0 ∥ L 2 for T + < t < T + + ε with some ε > 0 suﬃcien tly small. Hence the explicit formula abov e fails to yield a unitary map on L 2 + righ t after the breaking time T + > 0. In view of the sta- bilit y theorem ab ov e, this means that the discrete semigroup { A j t } j ∈ N with the con traction A t = e − itT cos θ S ∗ = e − i 2 t ( S + S ∗ ) S ∗ is not strongly stable for T + < t < T + + ε . F rom an op erator-analytic p oin t of view, it would b e in teresting to pro v e this failure of strong stabilit y for A t directly , without resorting to the shock formation abov e. In view of the preceding discussion, it is a remark able fact that we can rule out formation of sho c ks for (HWM) in contrast to the zero-disp ersion limit for (BO). As a matter of fact, the deep er reason lies in the entirely diﬀeren t spectral prop erties of the T oeplitz op erators T u 0 and T U 0 . More precisely , for all non- constan t u 0 ∈ C ( T ; R ) and an y U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )), it holds that: • T u 0 has no p oint sp ectrum σ p ( T u 0 ) = ∅ but only purely a.c.-spectrum. • T U 0 has only pure p oint spectrum σ ( T U 0 ) = σ p ( T U 0 ). In particular, these sp ectral properties of T U 0 allo w us to establish the unitarit y of the map U ( t ) in the explicit formula for (HWM). 4. Notation and preliminaries 4.1. V ector-v alued Hardy spaces and shift op erators. W e recall some basic facts and notions from the theory of L 2 -based Hardy spaces of functions taking v alues in a given Hilb ert space, also referred to as vector-v alued Hardy spaces. The in terested reader ma y consult [ 26, 24 ] for more bac kground. Let E be a complex (not necessarily separable) Hilb ert space with inner product ⟨·|·⟩ E and corresp onding norm | · | E . W e denote by L 2 ( T ; E ) the Hilb ert space HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 13 of strongly measurable and square-integrable functions on T v alued in E with the inner pro duct ⟨ F | G ⟩ = 1 2 π Z T ⟨ F ( θ ) | G ( θ ) ⟩ E dθ . By Parsev al’s theorem, w e notice that L 2 ( T ; E ) =  F = ∞ X n = −∞ b F n e inθ : ∞ X n = −∞ | b F n | 2 E < + ∞  , where b F n = 1 2 π R 2 π 0 F ( θ ) e − inθ dθ ∈ E denotes the n -th F ourier coeﬃcient of F . Next, we let Π denote the Cauc h y–Szeg˝ o pro jection on to the closed subspace of elemen ts in L 2 ( T ; E ) whose negative F ourier co eﬃcien ts v anish, that is, w e set Π ∞ X n = −∞ b F n e int ! = ∞ X n =0 b F n e int . The range of Π is the L 2 -based Hardy space of functions v alued in E such that L 2 + ( T ; E ) = Π( L 2 ( T ; E )) =  F = ∞ X n =0 b F n e int : b F n ∈ E , ∞ X n =0 | b F n | 2 E < ∞  , whic h is of course a Hilb ert space in its o wn right. Note that Π is an orthogonal pro jection and we denote b y Π − = I − Π its complementing orthogonal pro jection with range L 2 − ( T ; E ) = Π − ( L 2 ( T ; E ) =  F = X n< 0 b F n e int : b F n ∈ E , X n< 0 | b F n | 2 E < ∞  . T o streamline our notation, w e will henceforth often use the shorthand notation L 2 = L 2 ( T ; E ) and L 2 ± = L 2 ± ( T ; E ) whenev er the c hoice of E is clear from the con text. It is a classical fact that elements F ∈ L 2 + ( T ; E ) can b e iden tiﬁed with holomor- phic functions F : D → E such that sup r< 1 Z 2 π 0 | F ( r e iθ ) | 2 E dθ < + ∞ , in whic h case the equality ∥ F ∥ 2 L 2 = 1 2 π sup r< 1 R 2 π 0 | F ( r e iθ ) | 2 E dθ holds and the radial limit lim r → 1 − F ( r e iθ ) exists for a.e. θ ∈ T . By adapting common practice in the theory of Hardy spaces, we will freely mak e use of this canonical identiﬁcation of F ∈ L 2 + ( T ; E ) with the corresp onding holomorphic function F : D → E satisfying the condition stated ab ov e. On the Hardy space L 2 + , there is a canonical op erator S ∈ L ( L 2 + ) together with its adjoint S ∗ ∈ L ( L 2 + ), which are giv en by S F = S ∞ X n =0 b F n e inθ ! := ∞ X n =1 b F n − 1 e inθ , S ∗ F = S ∗ ∞ X n =0 b F n e inθ ! = ∞ X n =0 b F n +1 e inθ . The b ounded op erators S and S ∗ are usually referred to as the forwar d shift (or righ t shift) and the b ackwar d shift (or left shift) on L 2 + , resp ectively . W e easily note 14 P A TRICK G ´ ERARD AND ENNO LENZMANN that, for any F ∈ L 2 + when regarded as a holomorphic function F = F ( z ) with z ∈ D , w e can equiv alently write ( S F )( z ) = z F ( z ) , ( S ∗ F )( z ) = F ( z ) − F (0) z . In particular, the backw ard shift S ∗ will pla y a central role in the analysis of (HWM) on T . Moreov er, we recall the fundamen tal identities S ∗ S F = F and S S ∗ F = F − M ( F ) , (4.1) where M ( F ) = 1 2 π Z 2 π 0 F (e iθ ) dθ = b F 0 . is the orthogonal pro jection on to the subspace of constant functions E ⊂ L 2 + ( T ; E ) corresp onding to taking the mean (or equiv alen tly the 0-th F ourier coeﬃcient) of F ∈ L 2 + . Since M (( S ∗ ) k F ) = b F k for all k ∈ N , a geometric series expansion yields the identit y F ( z ) = M  ( I − z S ∗ ) − 1 F  (4.2) v alid for all F ∈ L 2 + ( T ; E ) and z ∈ D . In fact, w e can regard (4.2) as a r epr o ducing kernel formula for the Hardy space L 2 + ( T ; E ). In our analysis of (HWM), w e will mostly consider the Hilbert space E = C d × d of the complex d × d -matrices equipped with the standard (F rob enius) inner product ⟨ A, B ⟩ E = T r( AB ∗ ) = d X j,k =1 A j k B j k together with its corresp onding norm | A | E = ⟨ A, A ⟩ 1 / 2 E for matrices A ∈ C d × d . 4.2. T o eplitz and Hank el op erators. Recall that E is a complex Hilb ert space. Let L ( E ) denote the Banac h space of b ounded linear op erators on E into itself equipp ed with the op erator norm. W e use L ∞ ( T ; L ( E )) to denote the Banach space of (Bo chner) measurable maps from T to L ( E ) with ﬁnite norm giv en by ∥ U ∥ L ∞ := ess sup θ ∈ T ∥ U ( θ ) ∥ L ( E ) . Again, we remark that we will b e mainly in terested in the c hoice E = C d × d . Note that every U ∈ C d × d can b e naturally seen as an element in L ( E ) b y acting on elemen ts on E = C d × d b y left matrix multiplication. In particular, we see that L ∞ ( T ; C d × d ) ⊂ L ∞ ( T ; L ( E )) when E = C d × d . F or giv en U ∈ L ∞ ( T ; E ), we deﬁne the corresp onding T o eplitz op er ator with the sym b ol U by setting T U : L 2 + ( T ; E ) → L 2 + ( T ; E ) , F 7→ T U F := Π( UF ) . Lik ewise, w e deﬁne the Hankel op er ator with sym b ol U to be the op erator H U : L 2 + ( T ; E ) → L 2 − ( T ; E ) , F 7→ H U F := Π − ( UF ) . As an aside, we inform the reader that there exist alternative (but ultimately equiv- alen t) wa ys of deﬁning Hank el op erators in the literature; e.g., as the anti-linear op erator F 7→ Π( U F ) or b y using the unitary ﬂip op erator J on L 2 with J Π − = Π J . Ho wev er, we prefer the deﬁnition of H U ab o ve, as it turns out to b e the most con- v enient one for the analysis of (HWM) below. In particular, w e follo w the deﬁnition of Hankel operators as done in [ 24 ]. HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 15 Eviden tly , w e ha ve the op erator norm bounds with ∥ T U ∥ ≤ ∥ U ∥ L ∞ and ∥ H U ∥ ≤ ∥ U ∥ L ∞ . F urthermore, it is readily chec ked that the adjoints of T ∗ U : L 2 + → L 2 + and H ∗ U : L 2 − → L 2 + are found to b e T ∗ U f = Π( U ∗ f ) and H ∗ U f = Π( U ∗ f ), where U ∗ denotes the p oin twise adjoin t of U : T → L ( E ). In particular, w e see that T U = T ∗ U is self-adjoint if and only if U ( θ ) = U ( θ ) ∗ is self-adjoint on E for a.e. θ ∈ T . In addition, w e record t wo imp ortant facts ab out Hank el operators H U with sym b ol U ∈ L ∞ ( T ; E ) that will b e detailed and used in Section 5 b elo w. • H U is Hilb ert–Schmidt if and only if P n< 0 | n |∥ b U n ∥ 2 L ( E ) < ∞ . In the case E = C d × d , the latter condition means that Π − U belongs to the Sob olev space H 1 / 2 ( T ; C d × d ). • H U has ﬁnite r ank if and only if Π − U is a rational function of e iθ with θ ∈ T with no poles on the unit circle. Next, w e recall the following commutator formula for T o eplitz operators T U with the backw ard shift S ∗ . Lemma 4.1. F or U ∈ L ∞ ( T ; L ( E )) and F ∈ L 2 + ( T ; E ) , it holds [ S ∗ , T U ] F = S ∗ T U ( M ( F )) . In p articular, if E = C d × d and U ∈ L ∞ ( T ; C d × d ) , the identity ab ove r e ads [ S ∗ , T U ] F = ( S ∗ Π U ) M ( F ) , with the b ackwar d shift S ∗ acting on Π U ∈ L 2 + ( T , C d × d ) . R emark. As a direct consequence of Lemma 4.1, w e deduce that eigenspaces of T U are ne a rly S ∗ -invariant , whic h will turn out to be a k ey feature; see Section 5 b elo w. Pr o of. The proof of Lemma 4.1 is elementary , y et we pro vide the details for the reader’s con venience. Let F ∈ L 2 + ( T ; E ) be given. Using the iden tity S ∗ T U S = T U (whic h is straigh tforward to c heck for an y T oeplitz op erator T U ) in combination with identit y (4.1), we ﬁnd [ S ∗ , T U ] F = S ∗ T U F − T U S ∗ F = S ∗ T U F − S ∗ T U S S ∗ F = S ∗ T U ( M ( F )) . This sho ws the claimed identit y . If E = C d × d and U ∈ L ∞ ( T ; C d × d ), w e can write this identit y as stated ab ov e, suing that M ( F ) ∈ C d × d is a constant matix. □ F or later use, we also record the following key iden tity relating T oeplitz and Hank el operators: T FG − T F T G = H ∗ F ∗ H G (4.3) v alid for an y F , G ∈ L ∞ ( T ; L ( E )). The elementary pro of of (4.3) is left to the reader. 4.3. Sob olev-t yp e spaces and density of rational maps. In what follo ws, we tak e E = C d × d with some ﬁxed in teger d ≥ 2 equipped with standard (F rob enius) inner product denoted by ⟨ A | B ⟩ E = tr( AB ∗ ) for A, B ∈ C d × d . F or real s ≥ 0, w e deﬁne the Sob olev space for matrix-v alued functions on T giv en by H s := H s ( T ; C d × d ) := { F ∈ L 2 ( T ; C d × d ) : ∥ F ∥ H s < + ∞} with the norm given b y ∥ F ∥ 2 H s := P ∞ n = −∞ (1 + | n | 2 s ) | b F n | 2 F , and we set H ∞ := T s ≥ 0 H s ( T ). W e also deﬁne the family of Hardy–Sob olev spaces denoted b y H s + := H s ∩ L 2 + for s ≥ 0 together with the obvious deﬁnition of the space H ∞ + . 16 P A TRICK G ´ ERARD AND ENNO LENZMANN F or integers d ≥ 2 and 0 ≤ k ≤ d , we recall that the c omplex Gr assmannian Gr k ( C d ) are deﬁned as the set of matrices such that Gr k ( C d ) :=  U ∈ C d × d : U = U ∗ , U 2 = 1 d , tr( U ) = d − 2 k  , where 1 d denote the d × d –iden tity matrix. W e remind the reader that the trivial cases Gr 0 ( C d ) = { 1 d } and Gr d ( C d ) = {− 1 d } will b e mostly excluded in our discus- sion, i.e., we assume that 1 ≤ k ≤ d − 1 holds. F or real s ≥ 0, we in tro duce the space H s ( T ; Gr k ( C d )) :=  U ∈ H s ( T ; C d × d )) : U ( θ ) ∈ Gr k ( C d ) for a.e. θ ∈ T  with the ob vious deﬁnition H ∞ ( T ; Gr k ( C d )) := T s ≥ 0 H s ( T ; Gr k ( C d )). F urthermore, w e recall that R at ( T ; Gr k ( C d )) ⊂ H ∞ ( T ; Gr k ( C d )) denotes the set of rational maps U : T → Gr k ( C d ), i.e., each matrix entry of U is a rational function of z = e iθ ∈ ∂ D ∼ = T . Since Gr k ( C d ) is a smooth compact manifold without b oundary , the seminal w ork of Brezis–Niren berg [ 4 ] tells us that the space of smo oth maps C ∞ ( T ; Gr k ( C d )) is dense in H 1 / 2 ( T ; Gr k ( C d )). In fact, it w as sho wn in [ 13 ][Theorem A.2], that w e can restrict to rational maps to already obtain a dense subset. Theorem 4.1 (Density of rational maps) . F or any d ≥ 2 and 0 ≤ k ≤ d , the set R at ( T ; Gr k ( C d )) is dense in H 1 / 2 ( T ; Gr k ( C d )) . 5. Sp ectral analysis of T U In this section, w e deriv e some fundamental sp ectral prop erties of the T oeplitz op erator T U : L 2 + ( T , C d × d ) → L 2 + ( T ; C d × d ) , F 7→ T U ( F ) = Π( UF ) , for a given initial datum U ∈ H 1 / 2 ( T ; Gr k ( C d )) . Here and throughout the following, w e take d ≥ 2 and 1 ≤ k ≤ d − 1 to be ﬁxed in tegers. Since U ∗ = U and U 2 = 1 d holds a.e. on T holds, w e immediately see that T ∗ U = T U is self-adjoint and that T U is a contraction, i.e., w e hav e ∥ T U ∥ ≤ ∥ U ∥ L ∞ = 1. In fact, we will see b elo w that equalit y ∥ T U ∥ = 1 holds. 5.1. Key sp ectral prop erties. Our starting p oint is the follo wing key iden tity for the T oeplitz op erator T U , where we recall that H U : L 2 + ( T , C d × d ) → L 2 − ( T , C d × d ) , F 7→ H U ( F ) = Π − ( UF ) denotes the Hank el op erator with the matrix-v alued symbol U ∈ H 1 / 2 ( T ; Gr k ( C d )). As usual, we often use the shorthand notation L 2 + = L 2 + ( T ; C d × d ) in what follows. Lemma 5.1 (Key identit y) . We have the identity T 2 U = I − K U with K U = H ∗ U H U . Her e the op er ator K U : L 2 + → L 2 + is self-adjoint and tr ac e-class with T r( K U ) = T r( H ∗ U H U ) = 1 2 ∥ U ∥ 2 ˙ H 1 / 2 . HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 17 R emark. F rom the identit y T 2 U = I − K U and the compactness of K U , we see that T U is F redholm with ind( T U ) = 0 b y the self-adjointness of T U . F urthermore, w e see that 0 ∈ σ ( K U ) by compactness of K U , whence it follows that equality ∥ T U ∥ = 1 holds. Pr o of. The claimed iden tity follows directly from (4.3) using that U 2 = 1 d and U = U ∗ almost everywhere on T . T o sho w that the self-adjoin t op erator K U = H ∗ U H U is trace-class, we deﬁne E n,l = e inθ E l with n ≥ 0, where the matrices E l with l = 1 , . . . , d 2 form an orthonormal basis of C d × d . Thus the family ( E n,l ) n ≥ 0 , 1 ≤ l ≤ d 2 forms an orthonormal basis of L 2 + ( T ; C d × d ). W e calculate T r( H ∗ U H U ) = ∞ X n =0 d 2 X l =1 ⟨ H U E n,l | H U E n,l ⟩ = ∞ X n =0 d 2 X l =1 ⟨ Π − ( UE n,l ) | Π − ( UE n,l ) ⟩ = ∞ X n =0 d 2 X l =1 * X k + n< 0 b U k e i ( k + n ) θ E l | X m + n< 0 b U m e i ( m + n ) θ E l + = ∞ X n =0 X n + k< 0 | b U k | 2 E = ∞ X k< 0 | k || b U k | 2 E = ∥ Π − U ∥ 2 ˙ H 1 / 2 = 1 2 ∥ U ∥ 2 ˙ H 1 / 2 . In the last step, we used that U = U ∗ a.e. on T , which is easily seen to imply that ( b U k ) ∗ = b U − k for k ∈ Z and th us ∥ Π − U ∥ 2 ˙ H 1 / 2 = ∥ Π + U ∥ 2 ˙ H 1 / 2 = 1 2 ∥ U ∥ 2 ˙ H 1 / 2 . □ Let σ ( T U ) denote the sp ectrum of T U . W e recall that the discr ete sp e ctrum σ d ( T U ) and essential sp e ctrum σ e ( T U ) are given b y σ d ( T U ) = { µ ∈ σ ( T U ) : µ is an isolated eigen v alue with ﬁnite multiplicit y } , σ e ( T U ) = { µ ∈ σ ( T U ) : T U − µI is not F redholm } . Note that we do not need to distinguish here b et ween geometric and algebraic m ultiplicity b ecause T U = T ∗ U is self-adjoint. Also, by self-adjointness of T U , we ha ve that σ d ( T U ) = σ ( T U ) \ σ e ( T U ). Lemma 5.2. We have the fol lowing pr op erties. (i) The discr ete sp e ctrum of T U is given by σ d ( T U ) = { µ ∈ R : λ = 1 − µ 2 > 0 is an eigenvalue of K U } and σ d ( T U ) ⊂ ( − 1 , +1) is at most c ountable. (ii) The essential sp e ctrum of T U is given by σ e ( T U ) = {± 1 } . (iii) We have the sp e ctr al r epr esentation T U = X µ ∈ σ ( T U ) µP µ , wher e P µ denotes the ortho gonal pr oje ction onto Ker( T U − µI ) . As a c on- se quenc e, the sp e ctrum of T U is pur e p oint. Pr o of. Item (i) directly follo ws from the iden tity T 2 U = I − K U in Lemma 5.1 for the self-adjoint operator T U and by the compactness of the op erator K U . As for (ii), w e note that σ e ( T U ) ⊆ {± 1 } holds, since K U = I − T 2 U is compact. T o show the equality σ e ( T U ) = {± 1 } , w e argue as follo ws. Since 1 ≤ k ≤ d − 1, we 18 P A TRICK G ´ ERARD AND ENNO LENZMANN note that − 1 and +1 are b oth eigen v alues of the matrix U ( θ ) ∈ Gr k ( C d ) for a.e. T . Hence the matrix-v alued map G σ = U + µ 1 d ∈ L ∞ ( T ; C d × d ) is not inv ertible in L ∞ ( T ; C d × d ) for µ ∈ {± 1 } . F rom [ 24 ][Theorem 4.3, Chapter 3], we recall that T G with G ∈ L ∞ ( T ; C d × d ) is F redholm if and only if G − 1 ∈ L ∞ ( T ; C d × d ). Hence we conclude that T G = T U + µ 1 d = T U + µI is not F redholm for µ ∈ {± 1 } . This shows that {± 1 } ⊆ σ e ( T U ), which completes the pro of of σ e ( T U ) = {± 1 } . Finally , we remark that (iii) follows from the sp ectral theorem applied to the b ounded self-adjoin t op erator T U . □ 5.2. On inv ariant subspaces. As b efore, we assume that U ∈ H 1 / 2 ( T ; Gr k ( C d )) in what follows. By the commutator formula in Lemma 4.1, we deduce that an y eigenspace E µ ( T U ) = Ker( T U − µI ) m ust b e ne arly S ∗ -invariant ; see [ 25 ]. That is, w e ha v e the implication: F ∈ E µ ( T U ) and M ( F ) = 0 ⇒ S ∗ F ∈ E µ ( T U ) . This simple observ ation will turn out to be v aluable in our analysis further b elo w. Moreo ver, w e can naturally identify a closed subspace in L 2 + asso ciated to T U whic h is also S ∗ -in v ariant. Indeed, le t us deﬁne the closed subspace H := Ran( K U ) = Ran( H ∗ U H U ) = Ran( H ∗ U ) . This induces the orthogonal decomposition L 2 + = H ⊕ H ⊥ with H = Ran( H ∗ U ) and H ⊥ = Ker( H U ) . By a w ell-known result (see, e.g. [ 24 ]), the k ernel of a Hank el op erator is in v ariant under the forward shift S . Therefore we hav e S ( H ⊥ ) ⊂ H ⊥ , whic h implies that S ∗ ( H ) ⊂ H b y taking the adjoin t. On the other hand, since H is the closed linear span of the eigenfunctions of K U with positive eigenv alues λ > 0, we easily infer that T U ( H ) ⊂ H from Lemma 5.1 ab o ve. Hence we obtain the follo wing fact. Lemma 5.3. The close d subsp ac e H = Ran( K U ) is invariant under T U and S ∗ , i.e., we have T U ( H ) ⊂ H and S ∗ ( H ) ⊂ H . R emark. If H ⊊ L 2 + is a prop er subspace, which means that Ker( H U )  = { 0 } is non trivial, then by w ell-known Beurling–L ax–Halmos the or em it follo ws that H is a so-called mo del sp ac e ; see e.g. [ 23 ][Corollary I.6]. That is, w e can write H = (Θ L 2 + ( T ; V )) ⊥ with some subspace V ⊆ C d × d and some Θ ∈ L ∞ + ( T ; L ( V ; C d × d )) such that Θ( θ ) ∗ Θ( θ ) = I V for a.e. θ ∈ T . W e say that Θ is a left inner function . In the case of equalit y V = C d × d , we sa y that Θ is a two-side d inner function . How ever, we will not mak e further use of the this mo del space p oin t of view in our analysis of (HWM). With the help of the celebrated Kroneck er theorem for Hank el op erators, we can no w easily iden tify the case when H happens to ﬁnite-dimensional. Lemma 5.4 (` a la Kroneck er) . U ∈ R at ( T ; Gr k ( C d )) if and only if dim H < ∞ . R emark. In view of Lemma 5.2, we notice that dim H is ﬁnite-dimensional if and only if the discrete spectrum σ d ( T U ) is ﬁnite. HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 19 Pr o of. By Kronec ker’s theorem (see, e, g., [ 24 ][Chapter 2, Theorem 5.3], the Hankel op erator H ∗ V : L 2 − → L 2 + with V ∈ L ∞ ( T ; C d × d )) has ﬁnite rank if and only if its co-analytic part Π − V ( z ) = P n< 0 b V n z − n is a rational function in z ∈ D . On the other hand, since U = U ∗ almost everywhere on T , w e deduce that Π U = (Π − ( U ∗ )) ∗ + M ( U ) = (Π − U ) ∗ + M ( U ) with the constant function M ( U ) = b U 0 . This sho ws that Π + ( U ) is rational if and only if Π − ( U ) is rational. Therefore, we ha ve demonstrated that U = Π( U ) + Π − ( U ) is rational if and only if H is ﬁnite-dimensional. □ 6. Lax pair structure W e no w elab orate on the Lax pair structure for (HWM) posed on the torus T . Informed by our previous w ork [ 13 ] for the half-wa ve maps equation posed on the real line, we in tro duce the following un b ounded op erator acting on L 2 + with B U = − i 2 ( T U D + D T U ) + i 2 T | D | U (6.1) where D = − i∂ θ and U ∈ H s ( T ; Gr k ( C d )) with some s > 3 2 is given. F rom the discussion in Appendix A, w e deduce that B U with op erator domain Dom( B U ) = H 1 + is essential ly skew-adjoint . F or the rest of this subsection, w e will alwa ys make the following assumptions without further reference: • I ⊂ R is an in terv al with 0 ∈ I . • U ∈ C ( I ; H s ( T ; Gr k ( C d ))) with some s > 3 2 is a solution (HWM). W e refer the reader to App endix A b elo w, where we pro ve lo cal w ell-p osedness of (HWM) in H s with s > 3 / 2. W e deﬁne the op erators T U ( t ) and B U ( t ) for t ∈ I in an obvious manner. Also, from the discussion in App endix A, we infer that there exists a unique solution U ( t ) ∈ L ( L 2 + ) for t ∈ I solving the abstract op erator-v alued ev olution equation d dt U ( t ) = B U ( t ) U ( t ) for t ∈ I , U (0) = I . (6.2) Since B U ( t ) : H 1 + ⊂ L 2 + → L 2 + is essen tially skew-adjoin t, the solution U ( t ) is a unitary map on L 2 + for all t ∈ I . Moreo ver, w e ha ve U ( t ) : H 1 + → H 1 + for t ∈ I and U ( · ) F ∈ C 1 ( I ; L 2 + ) for an y giv en F ∈ H 1 + . By the essen tial skew-adjoin tness of B U ( t ) , it also suﬃces consider elemen ts in H 1 + in the calculations below. In what follo ws, w e will make use of these facts without further reference. W e are now ready to state the follo wing Lax pair prop ert y for suﬃcien tly regular solutions of (HWM). Lemma 6.1 (Lax pair structure) . We have the L ax evolution d dt T U ( t ) =  B U ( t ) , T U ( t )  for t ∈ I . As a c onse quenc e, we have the unitary e quivalenc e T U ( t ) = U ( t ) T U (0) U ( t ) ∗ for t ∈ I . R emark. Note that, by the regularity imp osed on U ( t ), it is easy to chec k that T U ( t ) preserv es the domain Dom( B U ( t ) ) = H 1 + . Hence the commutator ab ov e is 20 P A TRICK G ´ ERARD AND ENNO LENZMANN w ell-deﬁned on Dom( B U ( t ) ) for an y t ∈ I . Again, w e will mak e tacitly use of such facts b elo w. Pr o of. Fix t ∈ I and write U = U ( t ) for notational simplicity . First, w e note [ D , T U ] = T D U b y the Leibniz rule, whence it follows [ B U , T U ] = − i 2 [ T U D + D T U , T U ] + i 2 [ T | D | U , T U ] = − i 2 ( T U T D U + T D U T U ) + i 2 [ T | D | U , T U ] =: ( I ) + ( I I ) . Diﬀeren tiating the iden tity U 2 = 1 d on T , w e ﬁnd that ( D U ) U + U ( D U ) = 0. Th us T ( D U ) U + T ( U D ) U = 0 and by using the general identit y (4.3) w e get ( I ) = − i 2 ( T U T D U + T D U T U ) = − i 2 ( H ∗ U H D U − H ∗ D U H U ) = − i 2  H ∗ U H Π − ( D U ) − H ∗ Π( D U ) H U  . Here we also used that U = U ∗ on T and ( D U ) ∗ = − D U together with the facts H F = H Π − F and H ∗ G = H ∗ Π G . Similarly , by in voking (4.3) once again, w e deduce ( I I ) = i 2 [ T U , T | D | U ] = i 2  T [ U , | D | U ] − H ∗ U H | D | U + H ∗ | D | U H U  = i 2 T [ U , | D | U ] − i 2  H ∗ U H Π − ( | D | U ) − H ∗ Π( | D | U ) H U  . in view of U = U ∗ and ( | D | U ) ∗ = | D | U on T . Since Π − ( | D | U ) = − Π − ( D U ) and Π( D U ) = Π( | D | U ), we see that the parts containing the Hank el op erators cancel in the sum ab ov e. Therefore, [ B U , T U ] = ( I ) + ( I I ) = i 2 T [ U , | D | U ] = d dt T U , where the last equation directly follo ws from (HWM) itself. The identit y T U ( t ) = U ( t ) T U (0) U ( t ) ∗ for t ∈ I follows from the comm utator iden tity and (6.2). W e omit the details. This completes the pro of of Lemma 6.1. □ Recall that K U ( t ) = I − T 2 U ( t ) b y Lemma 5.1 with the trace-class op erator K U = H ∗ U H U : L 2 + → L 2 + . F rom Lemma 6.1 and the Leibniz rules for d dt and the commutator of operators, respectively , w e directly deduce the follo wing result. Corollary 6.1. As a c onse quenc e of L emma 6.1, we have d dt K U ( t ) =  B U ( t ) , K U ( t )  for t ∈ I and the unitary e quivalenc e K U ( t ) = U ( t ) K U (0) U ( t ) ∗ for t ∈ I . F or later use, let us also record another important feature of the Lax pair struc- ture. T o this end, we notice that we alwa ys hav e that C d × d ⊂ Ker( B U ( t ) ) holds, that is, B U ( t ) E = 0 for an y E ∈ C d × d ⊂ L 2 + . (6.3) Indeed, take a constant matrix F ∈ C d × d . Then, since D E = 0 and Π D = Π | D | , w e readily ﬁnd B U ( t ) E = − i 2 D Π( U ( t ) E ) + i 2 Π( | D | U ( t ) E ) = 0 . HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 21 Based on this simple observ ation, we can deduce that Π U ( t ) can b e simply ex- pressed in terms of the unitary map U ( t ) : L 2 + → L 2 + applied to the initial condition Π U (0). Lemma 6.2. The fol lowing identity holds true: Π U ( t ) = U ( t )Π U (0) for t ∈ I . Pr o of. Since U ( t ) : L 2 + → L 2 + is unitary , the claimed iden tity is equiv alen t to U ( t ) ∗ Π U ( t ) = Π U (0) . F urthermore, from U (0) ∗ = I , it suﬃces to show that the time deriv ativ e of left- hand side ab ov e is zero. Indeed, we ﬁnd d dt ( U ( t ) ∗ (Π U ( t )) = U ( t ) ∗  − B U ( t ) Π U ( t ) + ∂ t Π U ( t )  , (6.4) using the skew-symmetry of B U ( t ) . Next, b y the Lax equation in Lemma 6.1 and the fact B U ( t ) ( 1 d ) = 0 from ab o ve, w e obtain d dt T U ( t ) ( 1 d ) = B U ( t ) T U ( t ) ( 1 d ) − T U ( t ) B U ( t ) ( 1 d ) = B U ( t ) Π( U ( t )) . On the other hand, we eviden tly see that ∂ t Π U ( t ) = d dt T U ( t ) ( 1 ) and hence the righ t-hand side in (6.4) is equal to zero. This completes the proof. □ R emark. Since U ( t ) solv es (6.2), we deduce from Lemma 6.2 that ∂ t Π U ( t ) = B U ( t ) Π U ( t ) for t ∈ I . (6.5) W e remark that another (and more direct) w a y of obtaining this diﬀerential equa- tion follows from applying the Cauc hy–Szeg˝ o pro jection Π to (HWM) itself together with the general identit y Π( F G ) = Π( F Π( G )) + Π(Π( G ) F ) − Π( F )Π( G ) in com bination with the point wise identit y ( D U ) U + U ( D U ) = 0 b ecause of the p oin twise constrain t U 2 = 1 d . W e leav e the details to the in terested reader. 6.1. Explicit form ula. By further exploiting the Lax structure for (HWM), w e shall now ﬁnd an explicit form ula for the unitary propagator U ( t ) generated b y B U ( t ) in the setting of suﬃciently regular solutions, that is, w e alw ays assume that U ∈ C ( I ; H s ( T ; Gr k ( C d ))) is a solution of (HWM) with some s > 3 2 on some time in terv al I ⊂ R suc h that 0 ∈ I . W e start with some preliminaries, whic h will be essential for proving the explicit form ula stated in Theorem 6.1 b elow. W e begin by sho wing that the backshift op erator S ∗ enjo ys the follo wing unitary equiv alence on the time in terv al I . Lemma 6.3. F or any t ∈ I , it holds that U ( t ) ∗ S ∗ U ( t ) = e − itT U 0 S ∗ on L 2 + . Pr o of. W e divide the pro of in to tw o steps as follo ws. Step 1. Let t ∈ I b e given. W e ﬁrst pro ve the comm utator identit y [ S ∗ , B U ( t ) ] F = − iT U ( t ) S ∗ F for F ∈ H 1 + . (6.6) 22 P A TRICK G ´ ERARD AND ENNO LENZMANN Indeed, from the fact that [ S ∗ , D ] = S ∗ on H 1 + and Lemma 4.1, we ﬁnd [ S ∗ , T U D + D T U ] F = T U [ S ∗ , D ] F + [ S ∗ , T U ] D F + D [ S ∗ , T U ] F + [ S ∗ , D ] T U F = T U S ∗ F + ( S ∗ Π U ) M ( D F ) + D ( S ∗ Π U ) M ( F ) + S ∗ T U F = T U S ∗ F + S ∗ T U F + D ( S ∗ Π U ) M ( F ) , where we used that M ( D F ) = 0 for any F ∈ H 1 + . Next, we notice [ S ∗ , T | D | U ] F = ( S ∗ Π | D | U ) M ( F ) = ( S ∗ Π D U ) M ( F ) = ( S ∗ D Π U ) M ( F ) , since Π | D | = Π D = D Π. Recalling that B U = − i 2 ( T U D + D T U ) + i 2 T | D | U , w e deduce [ S ∗ , B U ] F = − i 2 ( T U S ∗ F + S ∗ T U F ) + i 2 [ S ∗ , D ](Π U ) M ( F ) = − i 2 ( T U S ∗ F + S ∗ T U F ) + i 2 ( S ∗ Π U ) M ( F ) = − iT U S ∗ F − i 2 [ S ∗ , T U ] F + i 2 ( S ∗ Π U ) M ( F ) = − iT U S ∗ F , using once again that [ S ∗ , T U ] F = ( S ∗ Π U ) M ( F ). This pro v es (6.6). Step 2. Let F ∈ H 1 + b e giv en. F rom (6.6) w e ﬁnd d dt U ( t ) ∗ S ∗ U ( t ) F = U ( t ) ∗ [ S ∗ , B U ( t ) ] U ( t ) F = U ( t ) ∗ ( − iT U ( t ) S ∗ ) U ( t ) F . Since T U ( t ) = U ( t ) T U (0) U ( t ) ∗ b y the Lax evolution in Lemma 6.1 and the fact that U ( t ) is unitary , we conclude d dt U ( t ) ∗ S ∗ U ( t ) F = − iT U (0) U ( t ) ∗ S ∗ U ( t ) F . By integration in time and using that U (0) = I , we infer U ( t ) ∗ S ∗ U ( t ) F = e − itT U 0 S ∗ F . By density , this iden tity extends to all of F ∈ L 2 + . The pro of of Lemma 6.3 is no w complete. □ The next result shows that the mean is preserv ed by the unitary maps U ( t ). Lemma 6.4. F or any F ∈ L 2 + and t ∈ I , we have M ( U ( t ) F ) = M ( F ) . Pr o of. Let F ∈ H 1 + and E ∈ C d × d b e a constant matrix. Using the sk ew-symmetry of B U ( t ) , we readily deduce that d dt ⟨ U ( t ) F | E ⟩ = ⟨ B U ( t ) U ( t ) F | E ⟩ = −⟨ U ( t ) | B U ( t ) E ⟩ = 0 thanks to (6.3). This shows that d dt ( U ( t ) F ) ⊥ C d × d and hence d dt M ( U ( t ) F ) = 0, whence it follo ws that M ( U ( t ) F ) = M ( F ) for all F ∈ H 1 + . Since M and U ( t ) are b ounded operators on L 2 + , this identit y extends to all F ∈ L 2 + b y densit y . □ W e are no w ready to establish the explicit form ula for (HWM) in the setting of suﬃcien tly smooth solutions on a giv en time interv al [0 , T ]. HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 23 Theorem 6.1 (Explicit formula in H s with s > 3 2 ) . L et U ∈ C ( I ; H s ( T ; Gr k ( C d ))) b e a solution of (HWM) with some s > 3 2 . Denote by { U ( t ) } t ∈ I the unitary maps on L 2 + as given by (6.4) . Then, for any F ∈ L 2 + , it holds that ( U ( t ) F )( z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 F  for ( t, z ) ∈ I × D . In p articular, we have that Π U ( t, z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  for ( t, z ) ∈ I × D . Pr o of. Let t ∈ I and F ∈ L 2 + b e given. F rom the repro ducing-k ernel formula (4.2) w e can write ( U ( t ) F )( z ) = M  ( I − z S ∗ ) − 1 U ( t ) F  for z ∈ D . No w, b y Lemma 6.3, we hav e S ∗ U ( t ) = U ( t )e − itT U 0 S ∗ . If w e tak e the geometric series expansion in z ∈ D , we deduce the identit y ( I − z S ∗ ) − 1 U ( t ) = U ( t )( I − z e − itT U 0 S ∗ ) − 1 for z ∈ D . Therefore, ( U ( t ) F )( z ) = M  U ( t )( I − z e − itT U 0 S ∗ ) − 1 F  = M  ( I − z e − itT U 0 S ∗ ) − 1 F  , where the last equation follo ws from Lemma 6.4 ab ov e. Finally , let us take F = Π U 0 ∈ L 2 + in the explicit formula derived abov e. In view of Lemma 6.2, w e conclude that Π U ( t, z ) = ( U ( t )Π U 0 )( z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  for any t ∈ I and z ∈ D . This completes the pro of of Theorem 6.1. □ 7. Global well-posedness for rational data In this section, we will use the explicit formula in Theorem 6.1 to sho w that rational initial data U 0 ∈ R at ( T ; Gr k ( C d )) will giv e rise to unique smo oth global-in- time solutions of (HWM). As a b y-pro duct of our analysis, we pro ve that rational solutions are in fact quasip erio dic in time, which implies a-priori b ounds on all Sob olev norms ∥ U ( t ) ∥ H s for any s > 0 in the rational case. 7.1. Global existence for rational data. Supp ose U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) is an initial datum for (HWM) – which is not necessarily rational for the moment. In view of the explicit formula stated in Theorem 6.1 v alid for suﬃciently smo oth solutions, we no w deﬁne the expression ( U ( t ) F )( z ) := M  ( I − z e − itT U 0 S ∗ ) − 1 F  for ( t, z ) ∈ R × D (7.1) for any given F ∈ L 2 + = L 2 + ( T ; C d × d ). Of course, the map U ( t ) dep ends on the initial datum U 0 . But for the sak e of simplicity w e hav e chosen to omit this dep endence in our notation. Let us b egin with a spectral result, whic h rules out the existence of eigenv alues of the contraction e − itT U 0 S ∗ : L 2 + → L 2 + on the unit circle. Recall that σ p ( T ) denotes the p oin t sp ectrum of an op erator T ∈ L ( H ) on a Hilb ert space H . Notice also here we do not assume that U 0 is rational for the follo wing result. Lemma 7.1. L et U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) b e given. F or al l t ∈ R , it holds that σ p (e − itT U 0 S ∗ ) ∩ ∂ D = ∅ . 24 P A TRICK G ´ ERARD AND ENNO LENZMANN R emark. F or a suﬃciently smooth solution U ∈ C ( I ; H s ) of (HWM), we can see that e − itT U 0 S ∗ for t ∈ I is unitarily equiv alen t to the backw ard shift S ∗ thanks to Lemma 6.3. Since it is w ell-known that σ p ( S ∗ ) ∩ ∂ D = ∅ , the assertion ab o ve immediately follows. How ever, the p oint here is that we do not assume the existence of a suﬃciently smooth solution but we pro vide a direct pro of of this claim. In fact, this result will turn out to be essen tial when extending rational solutions globally in time. Pr o of. W e argue b y contradiction. Assume there exists F ∈ L 2 + \ { 0 } suc h that e − itT U 0 S ∗ F = λ F (7.2) for some λ ∈ ∂ D . T aking the L 2 -norm on b oth sides while using that e − itT U 0 is unitary and | λ | = 1, we deduce that ∥ S ∗ F ∥ L 2 = ∥ F ∥ L 2 , which b y (4.1) implies M ( F ) = 0 . F rom Lemma 4.1 this yields that [ S ∗ , T U 0 ] F = 0. Thus b y applying T U 0 to (7.2) and using that T U 0 and e − itT U 0 comm ute, w e deduce e − itT U 0 S ∗ T U 0 F = λT U 0 F . Hence the eigenspace V λ := Ker(e − itT U 0 S ∗ − λI ) satisﬁes T U 0 ( V λ ) ⊂ V λ and therefore e itT U 0 ( V λ ) ⊂ V λ as well. Next, b y going bac k to (7.2), we infer that S ∗ F = λ e itT U 0 F ∈ V λ sho wing that S ∗ F ∈ V λ . Hence S ∗ F is also an eigenfunction of e − itT U 0 S ∗ with eigen v alue λ and w e deduce that M ( S ∗ F ) = 0. By iteration, it follows that M (( S ∗ ) n F ) = b F n = 0 for n = 0 , 1 , 2 , . . . But this shows that F = 0, whic h is the desired con tradiction. □ As a next step, w e introduce the closed subspace K in L 2 + b y setting K := H + C Π U 0 + C 1 d with H = Ran( H ∗ U 0 ) . (7.3) Recall that H is the closed subspace in tro duced in Section 5 ab o ve, whic h was found to b e inv arian t under b oth T U 0 and S ∗ . In general, it is easy to see that Π U 0 ∈ H . This fact forces us to in tro duce the slightly larger subspace K ab o ve. Prop osition 7.1. It holds that T U 0 ( K ) ⊂ K , S ∗ ( K ) ⊂ K , and Π U 0 ∈ K . Mor e over, we have dim K < ∞ if and only if U 0 ∈ R at ( T ; Gr k ( C d )) . Pr o of. F rom (7.3) it is eviden tly true that Π U 0 b elongs to K . Since S ∗ ( H ) ⊂ H and S ∗ ( 1 d ) = 0 and S ∗ Π U 0 = H ∗ U 0 (e − iθ ) ∈ Ran( H ∗ U 0 ) ⊂ K , we deduce that K is inv ariant under S ∗ . F urthermore, we note that T U 0 ( H ) ⊂ H and T U 0 ( 1 d ) = Π U 0 ∈ K as well as T U 0 (Π U 0 ) = Π( U 0 Π U 0 ) = U 2 0 − Π( U 0 ( I − Π) U 0 ) = 1 d − H ∗ U 0 (Π − U 0 ) ∈ C 1 d + H ⊂ K . This prov es that T U 0 ( K ) ⊂ K . Finally , it is clear that dim K < ∞ if and only if dim H < ∞ . But the latter statemen t is equiv alent to U 0 ∈ R at ( T ; Gr k ( C d )) thanks to Lemma 5.4. □ HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 25 By Prop osition 7.1, the subspace K is inv arian t under e − itT U 0 S ∗ . Because of Π U 0 ∈ K , it therefore suﬃces to consider the restrictions e − itT U 0 S ∗ | K and ( I − z e − itT U 0 S ∗ ) − 1 | K for z ∈ D in the expression for U ( t ) in tro duced abov e when applied to Π U 0 . Moreo v er, for rational initial data U 0 , the subspace K is ﬁnite-dimensional and our analysis simpliﬁes substantially , since the sp ectral result in Lemma 7.1 turns out to b e suﬃcien t for obtaining a-priori b ounds on all H s -norms uniformly in time. In fact, pro vided that U 0 is rational, we can deduce that the map t 7→ U ( t ) U 0 is quasi- p erio dic in t ∈ R yielding a-priori b ounds on any Sobolev norm. Lemma 7.2. L et U 0 ∈ R at ( T ; Gr k ( C d )) . Then the map t 7→ U ( t ) U 0 is quasi- p erio dic with r esp e ct to t ∈ R in the sense that ( U ( t ) U 0 )( z ) = G ( t λ , z ) for ( t, z ) ∈ R × D , with some inte ger N ≥ 1 , some c onstant λ = ( λ 1 , . . . , λ N ) ∈ R N , and some b ounde d and c ontinuous map G : T N × D → C d × d . In p articular, we have the a-priori b ounds sup t ∈ R ∥ U ( t ) U 0 ∥ H s ( T ) ≲ s, U 0 1 for any s ≥ 0 . R emarks. 1) The real num b ers { λ k } N k =1 are the eigen v alues of the restriction of T U 0 | K on to the ﬁnite-dimensional in v arian t subspace K . 2) Notice that the map t 7→ U ( t ) U 0 is p erio dic in t ∈ R if and only if all λ 1 , . . . , λ N lie on a line in Q . Pr o of. Let U 0 ∈ R at ( T ; Gr k ( C d )) b e given. Re call that dim K < ∞ b y Proposition 7.1. Since T U 0 ( K ) ⊂ K and by self-adjoin tness of T U 0 , we can write T U 0 | K = N X n =1 λ n P n for some 1 ≤ N ≤ dim K , where λ n ∈ R are the eigenv alues of T U 0 | K and P n : K → K denote the cor- resp onding orthogonal pro jections on to the eigenspaces E n = Ker( T U 0 | K − λ n I ) whic h satisfy E n ⊥ E m if n  = m . F or ω = ( ω 1 , . . . , ω N ) ∈ T N , let us introduce the linear map Ω( ω ) := N X n =1 e − iω n P n : K → K . Note that Ω( ω ) is unitary with Ω( ω ) ∗ = Ω( − ω ) and Ω( ω ) commutes with T U 0 | K . By a straightforw ard adaption of the pro of of Lemma 7.1 and noticing that S ∗ ( K ) ⊂ K , w e infer that σ (Ω( ω ) S ∗ | K ) ∩ ∂ D = ∅ for ω ∈ T N . Note also that σ (Ω( ω ) S ∗ | K ) = σ p (Ω( ω ) S ∗ | K ) since K is ﬁnite-dimensional. By compactness of T N , this implies that there is some constan t r < 1 such that σ (Ω( ω ) S ∗ | K ) ⊆ D r for ω ∈ T N , (7.4) where D r = { z ∈ C : | z | ≤ r } is the closed disc with radius r . Next, w e deﬁne the map G : T N × D → C d × d with G ( ω , z ) := M  ( I − z Ω( ω ) S ∗ | K ) − 1 Π U 0  . F rom (7.4) w e deduce that, for each ω ∈ T N , the map z 7→ G ( ω , z ) is a rational function in z whose p oles b elong to the set { z ∈ C : z − 1 ∈ C \ D r } ⊂ {| z | ≥ 26 P A TRICK G ´ ERARD AND ENNO LENZMANN r − 1 } . Thus z 7→ G ( ω , z ) extends smo othly to the b oundary ∂ D , whic h sho ws that G ( ω , · ) ∈ H s ( T ; C d × d ) for an y s ≥ 0. Since T N is compact and by contin uit y of the H s -norm of G ( ω , · ) with resp ect to ω , we obtain sup ω ∈ T N ∥ G ( ω , · ) ∥ H s ( T ) ≲ s, U 0 1 for any s ≥ 0 . (7.5) Finally , we observ e that e − itT U 0 | K = Ω( t λ ) with the v ector λ = ( λ 1 , . . . , λ n ) ∈ R N con taining the eigenv alues of T U 0 | K in tro- duced ab o ve. Therefore, ( U ( t ) U 0 )( z ) = M  ( I − z Ω( t λ ) S ∗ ) − 1 Π U 0  = G ( t λ , z ) for ( t, z ) ∈ R × D . In view of the previous discussion of G , we complete the pro of of Lemma 7.2. □ W e are now ready to derive the main result of this section dealing with rational initial data for (HWM). Corollary 7.1 (GWP for rational initial data) . F or every U 0 ∈ R at ( T ; Gr k ( C d )) , ther e exists a unique glob al-in-time solution U ∈ C ∞ ( R × T ) of (HWM) with initial datum U (0) = U 0 . In addition, the fol lowing pr op erties hold for al l t ∈ R : (i) A-priori b ound on H s : It holds that sup t ∈ R ∥ U ( t ) ∥ H s ≲ s, U 0 1 for al l s ≥ 0 . (ii) L ax evolution: Ther e exists a unitary map U ( t ) on L 2 + such that T U ( t ) = U ( t ) T U 0 U ( t ) ∗ , and we have the c onservation laws M ( U ( t )) = M ( U 0 ) , E [ U ( t )] = E [ U 0 ] for al l t ∈ R (iii) Explicit formula: F or any z ∈ D and t ∈ R , it holds that Π U ( t, z ) = ( U ( t )Π U 0 )( z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  . (iv) Pr eservation of r ationality: We have U ( t ) ∈ R at ( T ; Gr k ( C d )) for al l t ∈ R . Pr o of. Supp ose that U 0 ∈ R at ( T ; Gr k ( C d )) let U ∈ C ( I ; H s ( T ; Gr k ( C d ))) denote b e the corresp onding maximal solution giv en b y the local w ell-p osedness result in Lemma A.1 for initial data in H s with s > 3 2 . F rom Theorem 6.1 w e infer that Π U ( t, z ) = Φ( t )( U 0 )( z ) is giv en b y the explicit formula for t ∈ I and z ∈ D . But in view of Lemma 7.2 w e deduce that sup t ∈ I ∥ Π U ( t ) ∥ H s ( T ) < + ∞ , whic h implies that I = R m ust hold. Finally , w e remark that items (i)–(iii) directly follow from Lemma 6.1, Corollary 6.1, Theorem 6.1, and Lemma 7.2. T o see that (iv) holds true, we notice that Rank( K U ( t ) ) = Rank( K U 0 ) b y the Lax evolution in (i) and w e ha ve Rank( K U ) < ∞ if and only if U ∈ R at ( T ; Gr k ( C d )) in view of Lemma 5.4. □ HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 27 8. Stabilit y principle for explicit form ulae In this section, we study the explicit formula from more general op erator the- oretic p ersp ectiv e. Our key result will b e a general stability principle form ulated in Theorem 8.1 below, which can be applied to completely in tegrable PDEs in Hardy spaces L 2 + ( T ; E ) that feature explicit formulae such as the Benjamin–Ono equation, Calogero–Moser DNLS, and half-wa ve maps equation p osed on T . In our companion work, w e develop a corresp onding stability principle for explicit form ulae in volving the Hardy spaces L 2 + ( R ; E ), which arises for the corresp onding completely integrable PDEs on the real line. The main ideas developed in this section will sho w a close connection to the celebrated Wold de c omp osition for isometries on Hilb ert spaces; see e.g. [ 26, 23 ]. Ho wev er, our presentation will b e self-con tained. Thus it is (hop efully) accessible to readers without an y further operator-theoretic background. In particular, we will not take the W old decomp osition as given, but w e can rather derive this result as a corollary b elow. 8.1. Main result of this section. The goal of this subsection is to derive a general result for explicit form ulae that arise for the half-w av e maps equation on T as w ell as the Benjamin–Ono equation and the Calogero–Moser DNLS on T , together with their corresp onding zero-disp ersion limits. W e consider the follo wing general setting. Assume that E is a complex Hilbert space and let H = L 2 + ( T ; E ) be the corresp onding L 2 -based Hardy space of holo- morphic functions on D with v alues in E . As usual, we use S and S ∗ to denote the forw ard and bac kw ard shift on H , resp ectiv ely . Next, we suppose that L : Dom( L ) ⊂ H → H is a (p ossibly unbounded) self- adjoin t op erator. W e let { e − itL } t ∈ R denote the strongly con tinuous one-parameter unitary group on H generated b y L . Finally , for t ∈ R , we deﬁne the linear map U ( t ) : L 2 + ( T ; E ) → L 2 + ( T ; E ) giv en b y ( U ( t ) F )( z ) = M  ( I − z e − itL S ∗ ) − 1 F  with F ∈ L 2 + ( T ; E ) and z ∈ D . Recall that M ( G ) = 1 2 π R 2 π 0 G = b G 0 denotes the mean, which corresp onds to the orthogonal pro jection in L 2 + ( T ; E ) onto the subspace constant functions on T v alued in E . W e ha ve the following examples. • (HWM) on T : E = C d × d and L = T U 0 . • (CS-DNLS) on T : E = C and L = D ± T u 0 T u 0 . • (BO) on T : E = C and L = D − T u 0 . • Zero-disp ersion limit of (BO) on T : E = C and L = T u 0 . By the general discussion in Subsection 8.2 b elo w, it readily follo ws that U ( t ) is alwa ys a con traction, i.e., its operator norm is ∥ U ( t ) ∥ ≤ 1 for all t ∈ R . The follo wing general result now completely iden tiﬁes the case when U ( t ) is unitary . This insigh t will turn out to b e essen tial for our analysis of (HWM) with data in H 1 / 2 . Theorem 8.1 (Stabilit y principle) . The map U ( t ) : L 2 + ( T ; E ) → L 2 + ( T ; E ) is unitary for al l t ∈ R if and only if one of the fol lowing e quivalent c onditions hold. (i) Ker U ( t ) = { 0 } for al l t ∈ R . 28 P A TRICK G ´ ERARD AND ENNO LENZMANN (ii) lim j →∞ ∥ (e − itL S ∗ ) j F ∥ = 0 for al l F ∈ L 2 + ( T ; E ) and t ∈ R . In this c ase, the map t 7→ U ( t ) F b elongs to C ( R ; L 2 + ( T ; E )) for any F ∈ L 2 + ( T ; E ) . Before w e turn to the pro of of this theorem, we will ﬁrst deriv e some preliminary results in the next subsection. 8.2. Preliminaries for the pro of. Let H b e a complex Hilb ert space (not neces- sarily separable) with inner pro duct and norm denoted by ⟨·|·⟩ and ∥ · ∥ , resp ectiv ely . In the following discussion, w e alwa ys assume that Σ : H → H is an isometry , whic h by deﬁnition means that ∥ Σ F ∥ = ∥ F ∥ for each F ∈ H . In fact, the latter condition is easily seen to be equiv alent to Σ ∗ Σ = I , (8.1) where I denotes the iden tity on H . Let us deﬁne P Σ ∈ L ( H ) by setting P Σ := I − ΣΣ ∗ . In view of (8.1), w e chec k that P ∗ Σ = P Σ = P 2 Σ is an orthogonal pro jection on H on to E := Ker Σ ∗ , that is, we ha ve E = Ker Σ ∗ = Ran P and E ⊥ = Ran Σ = Ker P . Since E ⊂ H is a closed subspace, it is a Hilbert space in its o wn right with norm ∥ F ∥ E = ∥ F ∥ for F ∈ E . Let L 2 + ( T ; E ) denote the corresp onding Hardy space of E -v alued holomorphic functions G : D → E such that G = P ∞ n =0 G n z n with G n ∈ E and P ∞ n =0 ∥ G n ∥ 2 E < ∞ . T o simplify our notation, w e will sometimes write L 2 + instead of L 2 + ( T ; E ). F or the giv en isometry Σ : H → H , let us no w deﬁne the linear map U Σ : H → L 2 + ( T , E ) by setting ( U Σ F )( z ) := P Σ  ( I − z Σ ∗ ) − 1 F  (8.2) with F ∈ H and z ∈ D . Since ∥ Σ ∗ ∥ = ∥ Σ ∥ = 1 and thus σ (Σ ∗ ) ⊆ D , w e notice that I − z Σ ∗ has a b ounded inv erse for all z ∈ D . In fact, w e hav e the following result. Lemma 8.1. The line ar map U Σ : H → L 2 + ( T ; E ) is a c ontr action. F or al l F ∈ H , the fol lowing pr op erties hold. (i) We have the identity ∥ U Σ F ∥ 2 L 2 + = ∥ F ∥ 2 − lim N →∞ ∥ (Σ ∗ ) N F ∥ 2 . (ii) U Σ ( F ) = F whenever F ∈ E . (iii) The fol lowing intertwining r elations hold S U Σ = U Σ Σ and S ∗ U Σ = U Σ Σ ∗ on H . R emark. Since ∥ Σ ∗ ∥ = ∥ Σ ∥ = 1, the non-negativ e sequence ∥ F ∥ L 2 ≥ ∥ Σ ∗ F ∥ L 2 ≥ ∥ (Σ ∗ ) 2 F ∥ L 2 ≥ . . . ≥ 0 is monotone-decreasing. Hence the limit lim N →∞ ∥ (Σ ∗ ) N F ∥ = inf N ≥ 0 ∥ (Σ ∗ ) N F ∥ alw ays exists for any F ∈ H . HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 29 Pr o of. Let F ∈ H b e ﬁxed. F or any z ∈ D , w e can use the geometric series expansion to conclude ( U Σ F )( z ) = ∞ X n =0 ( P Σ (Σ ∗ ) n F ) z n . (8.3) Note that the series is locally uniformly conv ergent in D b ecause ∥ P Σ (Σ) ∗ F ∥ E ≤ ∥ (Σ ∗ ) n F ∥ ≤ ∥ F ∥ indep endent of n . Thus U Σ F : D → E is holomorphic on D and w e ha ve U Σ F ∈ L 2 + ( T , E ) if and only if ∥ U Σ F ∥ 2 L 2 + = ∞ X n =0 ∥ P Σ (Σ ∗ ) n F ∥ 2 E < ∞ . T o see that the series is ﬁnite, we recall that P Σ = I − ΣΣ ∗ , which implies that ∥ P Σ G ∥ 2 E = ∥ P Σ G ∥ 2 = ∥ G ∥ 2 − ∥ Σ ∗ G ∥ 2 for all G ∈ H . Applying this identit y to G = (Σ ∗ ) n F with n ∈ N , we ﬁnd ∥ P Σ (Σ ∗ ) n F ∥ 2 E = ∥ (Σ ∗ ) n F ∥ 2 − ∥ (Σ ∗ ) n +1 F ∥ 2 . (8.4) Therefore, we obtain the telescopic sum N − 1 X n =0 ∥ P Σ (Σ ∗ ) n F ∥ 2 E = ∥ F ∥ 2 − ∥ (Σ ∗ ) N F ∥ 2 for N ≥ 1 . This shows that ∥ U Σ F ∥ L 2 + ≤ ∥ F ∥ for all F ∈ H . Hence the linear map U Σ : H → L 2 + is well-deﬁned and, in fact, it is a contraction. By passing to the limit N → ∞ , w e deduce that ∥ U Σ F ∥ 2 L 2 + = ∥ F ∥ 2 − lim N →∞ ∥ (Σ ∗ ) N F ∥ 2 . This prov es item (i). Next, w e assume that F ∈ E = Ran P Σ = Ker Σ ∗ . Then (8.3) immediately yields that U Σ F = P Σ ( F ) = F whenever F ∈ E . This shows (ii). It remains to pro ve (iii). Recall that S denotes the forward shift on L 2 + ( T , E ). F or any F ∈ L 2 + ( T , E ) and z ∈ D , we th us ﬁnd S ( U Σ F )( z ) = ∞ X n =0 P Σ ((Σ ∗ ) n F ) z n +1 = ∞ X n =1 P Σ ((Σ ∗ ) n − 1 F ) z n = ∞ X n =1 P Σ ((Σ ∗ ) n Σ F ) z n = ∞ X n =0 P Σ ((Σ ∗ ) n Σ F ) z n = ( U Σ Σ F )( z ) , where we used that Σ ∗ Σ = I (since Σ is an isometry) and that P Σ (Σ F ) = 0. Lik ewise, for the bac kward shift S ∗ on L 2 + ( T , E ), we ﬁnd S ∗ ( U Σ F )( z ) = ∞ X n =0 P Σ ((Σ ∗ ) n +1 F ) z n = ∞ X n =0 P Σ ((Σ ∗ ) n Σ ∗ F ) z n = U Ω (Σ ∗ F )( z ) . This shows (iii) and completes the pro of of Lemma 8.1. □ As a next step, w e iden tify the case when U Σ happ ens to b e a unitary map. Lemma 8.2. The map U Σ : H → L 2 + ( T ; E ) is unitary if and only if one of the fol lowing e quivalent c onditions holds. (i) Ker U Σ = { 0 } . (ii) lim j →∞ ∥ (Σ ∗ ) j F ∥ = 0 for al l F ∈ H . 30 P A TRICK G ´ ERARD AND ENNO LENZMANN In this c ase, we have the unitary e quivalenc e U ∗ Σ S U Σ = Σ on H . Pr o of. W e divide the pro of into the following steps. Step 1. W e sho w that U Σ : H → L 2 + ( T ; E ) is unitary if and only if (i) holds. Clearly , the unitarity of U Σ implies (i). On the other hand, let us assume that Ker U Σ = { 0 } is trivial. Deﬁne the closed subspace X s := lin { Σ j E } ∞ j =0 . W e claim that U Σ is an isometry on X s , i.e., ∥ U Σ F ∥ L 2 + = ∥ F ∥ for all F ∈ X s . (8.5) T o see this, take any ﬁnite linear com bination F = P m j =0 Σ j E j with E j ∈ E . Then (Σ ∗ ) m +1 F = P m j =0 (Σ ∗ ) m +1 − j E j = 0 using that Σ ∗ Σ = I and (Σ ∗ ) k E = 0 for k ≥ 1. Thus w e see that lim N →∞ ∥ (Σ ∗ ) N F ∥ ≤ ∥ (Σ ∗ ) m +1 F ∥ = 0. F rom the identit y in Lemma 8.1 w e deduce that (8.5) holds on lin { Σ j E } ∞ j =0 , whic h b y densit y extends to its closure X s . This prov es (8.5). Next, we claim that U Σ : X s → L 2 + ( T ; E ) is surjective. Since isometries hav e closed range, it suﬃces to show U Σ ( X s ) is dense in L 2 + ( T ; E ). By Lemma 8.1, we ha ve E = U Σ ( E ) and, by iterating the intert wining identit y , w e obtain S j E = U Σ (Σ j E ) for j ≥ 1. Thus we obtain that S j E = z j E = U Σ (Σ j E ) ⊂ U Σ ( X s ) for all j ≥ 0 . Since the linear span of { z j E } ∞ j =0 is dense in L 2 + ( T ; E ), we conclude that U ( X s ) = L 2 + ( T ; E ). In summary , w e ha ve sho wn that U Σ : X s → L 2 + ( T ; E ) is unitary . It remains to show that H = X s . Since X s is closed, this follo ws if w e show that X ⊥ s = { 0 } . Indeed, let F ∈ X ⊥ s b e given. Then we necessarily hav e that F ⊥ Σ j E for j ≥ 0. Hence ⟨ F | Σ j E ⟩ = ⟨ (Σ ∗ ) j F | E ⟩ = 0 for j ≥ 0, which means that (Σ ∗ ) j F ∈ Ker P Σ = E ⊥ for j ≥ 0. But in view of (8.3) this implies that U Σ ( F ) = 0, whence it follows that X ⊥ s b elongs to Ker U Σ . Since Ker U Σ = { 0 } by assumption, this implies that X ⊥ s is trivial. This completes the pro of that condition (i) is also suﬃcien t for U Σ : H → L 2 + ( T ; E ) to b e unitary . Step 2. The implication (ii) ⇒ (i) is clear from Lemma 8.1. Assume now that (i) holds. Then U Σ : H → L 2 + ( T ; E ) is unitary by Step 1 ab o ve. By the identit y in Lemma 8.1, we deduce that (ii) must hold. Step 3. If U Σ : H → L 2 + ( T ; E ) is unitary , w e directly deduce that U ∗ Σ S U Σ = Σ on H from the in tertwining relation in Lemma 8.1. The pro of of Lemma 8.2 is no w complete. □ R emark. F or readers with an op erator-theoretic background, we remark that a straigh tforward generalization of the previous pro of yields the well-kno wn Wold de c omp osition for isometries on Hilbert spaces. More precisely , b y a slight extension of the pro of ab o ve, w e can deduce the unique orthogonal decomp osition H = X u ⊕ X s HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 31 with the following properties: • X u = Ker U Σ = { F ∈ H : ∥ (Σ ∗ ) j F = ∥ F ∥ for j ≥ 0 } • X s = lin { Σ j E } ∞ j =0 = { F ∈ H : lim j →∞ ∥ (Σ ∗ ) j F ∥ = 0 } . • U Σ : X s → L 2 + ( T ; E ) is unitary and U ∗ Σ S U Σ = Σ on X s . W e readily chec k that X u and X s are inv ariant under Σ and Σ ∗ , where Σ | X u is unitary and Σ | X s is unitarily equiv alent to right shift S on L 2 + ( T ; E ), i.e., Σ | X s is completely non-unitary (c.n.u.). This is the w ell-known W old decomp osition for the isometry Σ in H . Alternativ ely , we could hav e in vok ed the W old decomp osition for Σ and hav e w orked ‘backw ards’ to deduce Lemma 8.2 ab ov e. But since the map U Σ itself is our cen tral ob ject of in terest and to keep the discussion self-contained, we hav e c hosen not to tak e the W old decomp osition as given. F or later use, we also derive the following conv ergence result. Note that w e imp ose strong op erator con vergence on the adjoints. Lemma 8.3. L et Σ and Σ n with n ∈ N b e isometries in H such that Ker Σ ∗ n = Ker Σ ∗ = E for n ∈ N . If Σ ∗ n → Σ ∗ str ongly, then U Σ n → U Σ we akly. Pr o of. Since Ker Σ ∗ n = Ker Σ ∗ = E , we note that P Σ n = P Σ for all n ∈ N . Let F ∈ H b e giv en. W e need to show that U Σ n F ⇀ U Σ F in L 2 + ( T ; E ) . (8.6) Deﬁne the sequence G n := U Σ n F in L 2 + ( T ; E ). Note that ∥ G n ∥ L 2 + ≤ ∥ F ∥ b ecause U Σ n : H → L 2 + ( T ; E ) is a contraction. Up to passing to a subsequence, we can assume that G n ⇀ G in L 2 + ( T ; E ) with some limit G ∈ L 2 + ( T ; E ). W e claim that G ( z ) = ( U Σ F )( z ) for z ∈ D . By holomorphicit y on D , it suﬃces to show this claim on the smaller disk D 1 / 2 = { z ∈ C : | z | < 1 / 2 } . Indeed, we observ e ( U Σ n F )( z ) − ( U Σ F )( z ) = P Σ  ( I − z Σ ∗ n ) − 1 F − ( I − z Σ ∗ ) − 1 F  = P Σ  ( I − z Σ ∗ n ) − 1 ( z (Σ ∗ − Σ ∗ n ))( I − z Σ ∗ ) − 1 F  . Since σ (Σ ∗ n ) ⊆ D , we see that ∥ ( I − z Σ ∗ n ) − 1 ∥ ≤ 1 1 / 2 = 2 for z ∈ D 1 / 2 . Hence, for an y z ∈ D 1 / 2 , we ﬁnd ∥ ( U Σ n F )( z ) − ( U Σ F )( z ) ∥ E ≤ ∥ (Σ ∗ − Σ ∗ n )( I − z Σ ∗ ) − 1 F ∥ → 0 as n → ∞ , using that Σ ∗ n → Σ ∗ strongly as op erators on H . This pro ves that the w eak limit is given b y G = U Σ F . Since the limit is indep enden t of the c hosen subsequence, w e obtain that (8.6) holds. □ 8.3. Pro of of Theorem 8.1. W e consider the Hardy space H = L 2 + ( T ; E ), where E is some complex Hilbert space. F or t ∈ R , we deﬁne the isometry Σ( t ) := S e itL : H → H. Since Σ( t ) ∗ = e − itL S ∗ and e − itL is unitary , we readily chec k that Ker Σ( t ) ∗ = Ker S ∗ = E and that P Σ( t ) = I − Σ( t )Σ( t ) ∗ = I − S S ∗ = M for all t ∈ R . Hence we can inv ok e Lemma 8.2 to infer that U ( t ) : L 2 + ( T ; E ) → L 2 + ( T ; E ) is unitary if and only if one the equiv alent conditions (i) and (ii) in Theorem 8.1 holds. 32 P A TRICK G ´ ERARD AND ENNO LENZMANN Supp ose no w that U ( t ) : L 2 + ( T ; E ) → L 2 + ( T ; E ) is unitary for all t ∈ R . Let F ∈ L 2 + ( T ; E ) b e giv en and consider a sequence of times t n → t with some time t ∈ R . By the strong contin uity of the unitary group { e − itL } t ∈ R , we deduce that Σ( t n ) ∗ = e − it n L S ∗ → e − itL S ∗ = Σ( t ) ∗ strongly as operators. By Lemma 8.3, w e ha ve that U ( t n ) F ⇀ U ( t ) F in L 2 + ( T ; E ). Since the maps U ( t n ) and U ( t ) are unitary , we deduce that lim n →∞ ∥ U ( t n ) F ∥ = ∥ U ( t ) F ∥ = ∥ F ∥ , which implies that U ( t n ) F → U ( t ) F in L 2 + ( T ; E ). Hence the map t 7→ U ( t ) F b elongs to C ( R ; L 2 + ( T ; E )). This pro of of Theorem 8.1 is no w complete. □ 9. Global well-posedness in H 1 / 2 The goal of this section is to prov e global w ell-p osedness for (HWM) as stated in Theorem 2.1. W e organize the discussion as follo ws. 9.1. Extension to initial data in H 1 / 2 . Let us ﬁx in tegers d ≥ 2 and 1 ≤ k ≤ d − 1. Supp ose that U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) is an initial datum for (HWM). By Theorem 4.1, there is a sequence of rational initial data U 0 ,n ∈ R at ( T ; Gr k ( C d )) suc h that U 0 ,n → U 0 in H 1 / 2 . (9.1) Let U n = U n ( t ) ∈ C ( R ; H ∞ ) denote the unique global-in-time solutions of (HWM) with initial datum U n (0) = U 0 ,n as pro vided by Corollary 7.1 ab ov e. W e use the short-hand notation L 2 + = L 2 + ( T ; C d × d ) in what follows. F or t ∈ R , w e consider the maps U n ( t ) : L 2 + → L 2 + and U ( t ) : L 2 + → L 2 + with ( U n ( t ) F )( z ) = M  ( I − z e − itT U 0 ,n S ∗ ) − 1 F  , ( U ( t ) F )( z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 F  . W e record the follo wing facts v alid for any t ∈ R , sequences t n → t , and n ∈ N : (P 1 ) U n ( t ) : L 2 + → L 2 + is unitary . (P 2 ) Π U n ( t ) = U n ( t )Π U 0 ,n . (P 3 ) e − it n T U 0 ,n S ∗ → e − itT U 0 S ∗ in the strong op erator top ology . Note that (P 1 ) and (P 2 ) directly follow from Corollary 7.1. As for (P 3 ), we re- mark that the strong op erator con vergence T U 0 ,n → T U 0 follo ws from (9.1) and b y dominated conv ergence. By self-adjointness of T U 0 ,n and T U 0 together with T U 0 ,n → T U 0 strongly and t n → t , w e easily deduce (P 3 ). W e observe the following con vergence prop erties. Lemma 9.1. L et U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) . Supp ose that the se quenc e U 0 ,n ∈ R at ( T ; Gr k ( C d )) with n ∈ N satisﬁes U 0 ,n → U 0 in H 1 / 2 and let t n → t . Then U n ( t n )Π U 0 ,n → U ( t )Π U 0 in L 2 and U n ( t n )Π U 0 ,n ⇀ U ( t )Π U 0 in H 1 / 2 In p articular, the limit U ( t )Π U 0 is indep endent of the chosen se quenc e of r ational initial data U 0 ,n ∈ R at ( T ; Gr k ( C d )) . Pr o of. First, we notice that ∥ U n ( t n )Π U 0 ,n − U n ( t n )Π U 0 ∥ L 2 ≤ ∥ Π( U 0 ,n − U 0 ) ∥ L 2 → 0 (9.2) HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 33 using that ∥ U n ( t n ) ∥ ≤ 1 for all n ∈ N . F rom the strong op erator conv ergence (P 3 ) and Lemma 8.3, we deduce that U n ( t n )Π U 0 ⇀ U ( t )Π U 0 in L 2 + . T ogether with (9.2) this implies U n ( t n )Π U 0 ,n ⇀ U ( t )Π U 0 in L 2 . Eviden tly , the limit is indep endent of the chosen sequence U 0 ,n of rational data. Next, since U 0 ,n with n ∈ N forms a bounded sequence in H 1 / 2 and b y conserv ation la ws for the corresp onding solutions U n ∈ C ( R ; H ∞ ) in Corollary 7.1, w e use (P 2 ) to ﬁnd sup n ∈ N ∥ U n ( t n )Π U 0 ,n ∥ H 1 / 2 ≤ sup n ∈ N ∥ U 0 ,n ∥ H 1 / 2 < ∞ . By Rellic h compactness H 1 / 2 ( T ) ⊂ L 2 ( T ), we infer that U n ( t n )Π U 0 ,n → U ( t )Π U 0 in L 2 and U n ( t n )Π U 0 ,n ⇀ U ( t )Π U 0 in H 1 / 2 . □ As a next step, w e in tro duce the map Φ : R × R at ( T ; Gr k ( C d )) → R at ( T ; Gr k ( C d )) (9.3) b y setting ( t, U 0 ) 7→ Φ t ( U 0 ) := U ( t )Π U 0 + ( U ( t )Π U 0 ) ∗ − M ( U 0 ) . F or all U 0 ∈ R at ( T ; Gr k ( C d )), we note that Φ t ( U 0 ) ∈ C ( R ; H ∞ ) yields the unique global-in-time solution of (HWM) with rational initial datum Φ 0 ( U 0 ) = U 0 . Lemma 9.2. The map Φ deﬁne d in (9.3) admits a unique c ontinuous extension Φ : R × H 1 / 2 ( T ; Gr k ( C d )) → H 1 / 2 ( T ; Gr k ( C d )) . Mor e over, for any U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) , we have that U ( t ) = Φ t ( U 0 ) is a we ak solution of (HWM) with initial datum U (0) = U 0 and it holds U ( · ) ∈ C ( R ; L 2 ) ∩ C ( R ; H 1 / 2 w ) , wher e H 1 / 2 w denotes H 1 / 2 e quipp e d with the we ak top olo gy. In addition, M ( U ( t )) = M ( U 0 ) and E [ U ( t )] ≤ E [ U 0 ] for al l t ∈ R . Pr o of. W e take a sequence U 0 ,n ∈ R at ( T ; Gr k ( C d )) such that U 0 ,n → U 0 in H 1 / 2 and assume that t n → t . By Lemma 9.1 and the deﬁnition of Φ t , w e conclude that Φ t n ( U 0 ,n ) → Φ t ( U 0 ) in L 2 Φ t n ( U 0 ,n ) ⇀ Φ t ( U 0 ) in H 1 / 2 . (9.4) This sho ws that the map Φ initially deﬁned for rational initial data extends uniquely to initial data in H 1 / 2 ( T ; Gr k ( C d )) so that the map Φ : R × H 1 / 2 ( T ; Gr k ( C d )) → R × H 1 / 2 w ( T ; C d × d ) is contin uous, where H 1 / 2 w denotes H 1 / 2 equipp ed with the weak topology . Let us write U ( t ) = Φ t ( U 0 ). W e claim that U ( t, θ ) ∈ Gr k ( C d ) for all t ∈ R and a.e. θ ∈ T . Indeed, let us ﬁx t ∈ R . In view of (9.4), we conclude that U n ( t, θ ) → U ( t, θ ) in C d × d for almost every θ ∈ T with the smo oth rational solutions U n ∈ C ( R ; H ∞ ) of (HWM). This p oint wise con vergence implies that the limit U ( t ) b elongs to H 1 / 2 ( T ; Gr k ( C d )) for all t ∈ R . F rom the previous discussion w e see that U ( t ) = Φ t ( U 0 ) ∈ C ( R ; L 2 ) ∩ C ( R ; H 1 / 2 w ) . 34 P A TRICK G ´ ERARD AND ENNO LENZMANN By passing to the limit n → ∞ for the smo oth rational solutions U n ( t ) ∈ C ( R ; H ∞ ) and the fact that ∂ t U n is uniformly b ounded in C ( R ; H − 1 / 2 ), we readily deduce that passing to the limit n → ∞ yields that U ( t ) ∈ C ( R ; L 2 ) ∩ C ( R ; H 1 / 2 w is a w eak solution of (HWM) with initial datum U (0) = U 0 . Finally , from the conserv ation laws M ( U n ( t )) = M ( U 0 ,n ) and E [ U n ( t )] = E [ U 0 ,n ] for the smo oth rational solutions U n ∈ C ( R ; H ∞ ) together with (9.4) and the fact U 0 ,n → U 0 in H 1 / 2 , we easily deduce that M [ U ( t )] = M [ U 0 ] and E [ U ( t )] ≤ E [ U 0 ] for all t ∈ R , whic h ﬁnishes the proof. □ 9.2. Lax ev olution and energy conserv ation. At this point, it is conceiv able that the solution U ( t ) = Φ t ( U 0 ) obtained ab o ve ma y exhibit a loss of energy , i.e., E [ U ( t )] < E [ U 0 ] for some t ∈ R , whic h is tantamoun t to the failure of strong contin uity of the map t 7→ Φ t ( U 0 ) in H 1 / 2 . T o rule out this scenario, and thus pro ving energy conserv ation as a b y-pro duct, we will deriv e that U ( t ) : L 2 + → L 2 + is unitary for all t ∈ R and energy conserv ation will follow by a corresp onding Lax ev olution b y U ( t ) for the trace-class op erator K U ( t ) = I − T 2 U ( t ) with E [ U ( t )] = T r( K U ( t ) ). W e start with the following k ey observ ation concerning the transp ort of eigen- functions of T U 0 in terms of the map U ( t ). Lemma 9.3. L et t ∈ R b e given. Supp ose F ∈ L 2 + \ { 0 } is an eigenfunction for T U 0 with eigenvalue µ ∈ σ ( T U 0 ) . Then F ( t ) := U ( t ) F satisﬁes T U ( t ) F ( t ) = µ F ( t ) and F ( t )  = 0 . Pr o of. W e divide the pro of into the following steps. Step 1. As before, let U 0 ,n ∈ R at ( T ; Gr k ( C d )) b e a sequence of rational data with U 0 ,n → U 0 in H 1 / 2 . Likewise, we denote b y U n ( t ) the corresponding solutions C ( R ; H ∞ ) of (HWM) with U n (0) = U 0 ,n and w e let U n ( t ) : L 2 + → L 2 + denote the corresponding sequence of unitary maps. F rom the Lax evolution for rational solutions we infer that T U n ( t ) U n ( t ) F = U n ( t ) T U 0 F = µ U n ( t ) F (9.5) where F ∈ L 2 + \ { 0 } solves T U 0 F = µ F . Note that U n ( t ) F ⇀ U ( t ) F in L 2 analogous to the pro of of Lemma 9.1. Moreo ver, from U n ( t ) → U ( t ) in L 2 w e easily deduce that T U n ( t ) → T U ( t ) strongly as operators. By this fact and the self-adjoin tness of T U n ( t ) and T U ( t ) , it is elemen tary to c heck that T U n ( t ) U n ( t ) F ⇀ T U ( t ) U ( t ) F . By passing to the limit n → ∞ in (9.5), w e obtain that F ( t ) := U ( t ) F satisﬁes T U ( t ) F ( t ) = µ F ( t ) . Step 2. W e no w claim that F ( t )  = 0 . Recalling Lemma 8.1, we see that F ( t ) = U ( t ) F  = 0 pro vided that M ( F )  = 0. Hence it remains to discuss the case when M ( F ) = 0 for the rest of the pro of. HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 35 Supp ose no w that F  = 0 with M ( F ) = b F 0 = 0. Since F  = 0, we can deﬁne n := min { k ≥ 1 : M (( S ∗ ) k F ) = b F k  = 0 } . Next, we consider the eigenspace E µ = Ker( T U 0 − µI ) . As an eigenspace for the T o eplitz operator T U 0 , we recall that E µ m ust b e ne arly S ∗ -invariant , i.e., G ∈ E µ and M ( G ) = 0 ⇒ S ∗ G ∈ E µ , whic h is a direct consequence of the commutator iden tity in Lemma 4.1. Therefore, ( S ∗ ) j F ∈ E µ for j = 0 , . . . , n. No w w e recall the in tertwining iden tity from Lemma 8.1, which giv es us S ∗ U ( t ) = U ( t )e − itT U 0 S ∗ . (9.6) By iterating this identit y n times and applying it to the vector F , we obtain ( S ∗ ) n U ( t ) F = U ( t )(e − itT U 0 S ∗ ) n F . Since e − itT U 0 G = e − itµ G for G ∈ E µ , we ﬁnd (e − itT U 0 S ∗ ) n F = e − inµt ( S ∗ ) n F , whence it follows that ( S ∗ ) n U ( t ) F = e − inµt U ( t )( S ∗ ) n F  = 0 , using that M (( S ∗ ) n F )  = 0 and the fact that U ( t ) G  = 0 if M ( G )  = 0 b y Lemma 8.1. Thus w e deduce that F ( t ) = U ( t ) F  = 0 m ust hold. □ W e next establish U ( t ) : L 2 + → L 2 + has trivial k ernel for all t ∈ R , which pro ves that it is a unitary map thanks to Theorem 8.1. Lemma 9.4. It holds that Ker U ( t ) = { 0 } for al l t ∈ R . Pr o of. Let t ∈ R b e given. F or µ ∈ σ ( T U 0 ), we consider the eigenspaces E µ (0) = Ker( T U 0 − µI ) and E µ ( t ) = Ker( T U ( t ) − µI ) . By Lemma 9.3, we infer that U ( t ) : E µ (0) → E µ ( t ) is injective. Assume now that there exists some F ∈ L 2 + with F  = 0 suc h that U ( t ) F = 0. F rom Section 5 w e recall that σ ( T U 0 ) = σ p ( T U 0 ) is at most countable and only p oin t sp ectrum. By self-adjointness of T U 0 , we can write F = X µ ∈ σ ( T U 0 ) F µ with F µ ∈ E µ (0) . Since U ( t ) maps E µ (0) into E µ ( t ), we observ e that U ( t ) F = X µ ∈ σ ( T U 0 ) U ( t ) F µ with U ( t ) F µ ∈ E µ ( t ) . Because E µ ( t ) ⊥ E ν ( t ) whenever µ  = ν by self-adjoin tness of T U ( t ) , it follo ws from U ( t ) F = 0 that U ( t ) F µ = 0 for all µ ∈ σ ( T U 0 ) . 36 P A TRICK G ´ ERARD AND ENNO LENZMANN On the other hand, since F  = 0, there exists some eigenv alue µ ∗ ∈ σ ( T U 0 ) suc h that F µ ∗  = 0. Since U ( t ) : E µ ∗ (0) → E µ ∗ ( t ) is injectiv e, this contradicts that U ( t ) F µ ∗ = 0. □ With the aid of Lemma 9.4, w e are ready to establish the following cen tral result. Lemma 9.5 (Lax evolution and energy conserv ation) . F o r al l t ∈ R , the map U ( t ) : L 2 + → L 2 + is unitary and it holds that T U ( t ) = U ( t ) T U 0 U ( t ) ∗ . In p articular, the tr ac e-class op er ator K U ( t ) = I − T 2 U ( t ) satisﬁes K U ( t ) = U ( t ) K U 0 U ( t ) ∗ , and we have ener gy c onservation E [ U ( t )] = T r( K U ( t ) ) = T r( K U 0 ) = E [ U 0 ] for al l t ∈ R . Pr o of. That U ( t ) : L 2 + → L 2 + is unitary for all t ∈ R follows from Theorem 8.1 together with Lemma 9.4. T o pro ve the unitary equiv alence of T U ( t ) and T U 0 , let U 0 ,n ∈ R at ( R ; Gr k ( C d )) b e a sequence of rational initial data such that U 0 ,n → U 0 in H 1 / 2 . As usual, we denote b y U n ( t ) = Φ t ( U 0 ,n ) the corresp onding smo oth global-in-time solutions. By the Lax evolution, it holds that T U n ( t ) U n ( t ) = U n ( t ) T U 0 ,n with the corresponding unitary maps U n ( t ) : L 2 + → L 2 + for n ∈ N . Since U n ( t ) ⇀ U ( t ) and T U n ( t ) → T U ( t ) strongly for any t ∈ R , the self-adjointness of T U n ( t ) and T U ( t ) allo ws us to pass to the limit n → ∞ in the iden tity ab o ve to deduce that T U ( t ) U ( t ) = U ( t ) T U 0 . Thanks to the unitarit y of U ( t ) : L 2 + → L 2 + , w e infer that T U ( t ) = U ( t ) T U 0 U ( t ) ∗ . The unitary equiv alence K U ( t ) = U ( t ) K U 0 U ( t ) ∗ is now a trivial consequence of the identit y K U ( t ) = I − T 2 U ( t ) . Finally , w e recall from Lemma 5.1 that E [ U ( t )] = T r( K U ( t ) ). T ogether with the fact that T r( U K U ∗ ) = T r( K ) for unitary maps U and trace-class op erators K this completes the proof. □ 9.3. Pro of of Theorem 2.1. W e now ready to pro ve global well-posedness for (HWM) with initial data in H 1 / 2 as stated in Theorem 2.1. In fact, the pro of follo ws for directly from the preceding discussion. F or the reader’s con venience, we detail the arguments as follo ws. First, we sho w the contin uity of the map Φ : R × H 1 / 2 ( T ; Gr k ( C d )) → H 1 / 2 ( T ; Gr k ( C d )) , whic h w as obtained as a unique extension from rational initial data by Lemma 9.1. Recall that Φ is kno wn to be weakly con tinuous. Th us it remains to sho w that ∥ Φ t n ( U 0 ,n ) ∥ H 1 / 2 → ∥ Φ t ( U 0 ) ∥ H 1 / 2 whenev er ( t n , U 0 ,n ) → ( t, U 0 ) in R × H 1 / 2 ( T ; Gr k ( C d )). Indeed, using the fact that ∥ F ∥ 2 H 1 / 2 = | M ( F ) | 2 + 2 E [ F ] for F ∈ H 1 / 2 together with the conserv ation of mean and energy conserv ation by Lemma 9.1 and 9.5, we directly deduce that ∥ Φ t n ( U 0 ,n ) ∥ 2 H 1 / 2 = ∥ U 0 ,n ∥ 2 H 1 / 2 → ∥ U 0 ∥ 2 H 1 / 2 = ∥ Φ t ( U 0 ) ∥ 2 H 1 / 2 HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 37 pro vided that U 0 ,n → U 0 in H 1 / 2 . (i)–(ii). By the contin uity of Φ, w e deduce that U ∈ C ( R ; H 1 / 2 ( T ; Gr k ( C d ))) holds and from Lemma 9.1 we recall that U ( t ) = Φ t ( U 0 ) is a weak solution of (HWM) with U (0) = U 0 . Also, w e ha ve energy conserv ation E [ U ( t )] = E [ U 0 ] for all t ∈ R thanks to Lemma 9.5. (iii). Supp ose that U 0 ,n ∈ H 1 / 2 ( T ; Gr k ( C d )) is a sequence with U 0 ,n → U 0 in H 1 / 2 . F or any compact interv al I ⊂ R , we claim that sup t ∈ I ∥ Φ t ( U 0 ,n ) − Φ t ( U 0 ) ∥ H 1 / 2 → 0 as n → ∞ . W e argue b y con tradiction and assume that there exist a sequence t n ∈ I and some ε 0 > 0 suc h that ∥ Φ t n ( U 0 ,n ) − Φ t ( U 0 ) ∥ H 1 / 2 ≥ ε 0 for all n ∈ N . By compactness of I , we can assume that t n → t for some t ∈ I by passing to a subsequence if necessary . But since Φ t n ( U 0 ,n ) → Φ t ( U 0 ) in H 1 / 2 b y the contin uity of Φ, w e arrive at a contradiction. (iv). The group prop erty Φ t + s ( U 0 ) = Φ t (Φ s ( U 0 )) for t, s ∈ R holds for rational initial data U 0 ∈ R at ( T ; Gr k ( C d )) b y uniqueness of these solutions in the class C ( R ; H 1 / 2 ). By densit y of rational data and contin uity of Φ, w e extend the group prop ert y abov e to an y U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )). (v). W e recall from Corollary 7.1 that Φ t ( U 0 ) ∈ R at ( T ; Gr k ( C d )) holds for all t ∈ R pro vided that U 0 ∈ R at ( T ; Gr k ( C d )). Finally , let us suppose that e Φ : R × H 1 / 2 ( T ; Gr k ( C d )) → H 1 / 2 ( T ; Gr k ( C d )) is a contin uous map suc h that prop erties (i) and (v) hold. F or U 0 ∈ R at ( T ; Gr k ( C d )), this is seen to imply that e U ( t ) = e Φ t ( U 0 ) is a rational solution of (HWM) in C ( R ; H ∞ ). But by Corollary 7.1 it follows that Π e U ( t, z ) = M  ( I − z e − itT U 0 S ∗ ) − 1 Π U 0  is given by the explicit form ula. Th us Φ and e Φ coincide on rational initial data, and by the uniqueness of the extension to H 1 / 2 , we conclude that Φ and e Φ are iden tical. This pro ves uniqueness of the ﬂow map Φ. The pro of of Theorem 2.1 is no w complete. □ 10. Almost Periodicity 10.1. Preliminaries. F or the reader’s con venience, we ﬁrst collect some funda- men tal facts about almost p erio dic functions v alued in Banach spaces. Let X b e a Banach space and denote by BC( R ; X ) the space of b ounded and con tinuous functions from R to X , which is a Banac h space endow ed with the norm ∥ f ∥ BC = sup t ∈ R ∥ f ( t ) ∥ X . W e recall that f ∈ BC( R ; X ) is said to b e almost p erio dic (in the sense of Bohr) if for every ε > 0, there exists L > 0 such that every interv al I ⊂ R of length L con tains some τ ∈ I with sup t ∈ R ∥ f ( t + τ ) − f ( t ) ∥ X ≤ ε. 38 P A TRICK G ´ ERARD AND ENNO LENZMANN Recall that Bo chner’s criterion states that f ∈ BC( R ; X ) is almost p erio dic if and only if the set of all its translates T rans( f ) = { f ( · + a ) : a ∈ R } is relativ ely compact in the Banach space BC( R ; X ). In particular, w e see that the space of almost p erio dic functions AP( R ; X ) := { f ∈ BC( R ; X ) : f : R → X is almost p erio dic } is a Banach space in its own righ t, equipp ed with the sup-norm. 10.2. Almost p erio dicit y for (HWM). The goal of this subsection is to pro ve Φ t ( U 0 ) ∈ AP( R ; H 1 / 2 ) for all U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) for the solutions of (HWM) as provided by Theorem 2.1. Again, w e will make strong use of the explicit form ula for (HWM) as follo ws. As usual, we let d ≥ 2 and 1 ≤ k ≤ d − 1 b e ﬁxed in tegers and we supp ose that U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) is an initial datum for (HWM). F or notational conv enience, w e denote Ω( t ) := e − itT U 0 : L 2 + → L 2 + for t ∈ R with the Hardy space L 2 + = L 2 + ( T ; C d × d ). Using that the sp ectrum σ ( T U 0 ) is at most countable, we establish with the following compactness prop ert y for the strongly contin uous one-parameter unitary group { Ω( t ) } t ∈ R . Lemma 10.1. F or every se quenc e ( a n ) in R , ther e is a subse quenc e ( a ′ n ) such that Ω( a ′ n ) → Ω ∞ str ongly , wher e Ω ∞ is a unitary map in L 2 + . R emark. F or notational con venience, w e use ( a ′ n ) to denote subsequences of ( a n ), since the real n umbers a n app ear in Ω( a n ) = e − ia n T U 0 and so introducing further subindices in the exp onent w ould lead to clumsy notation. Pr o of. F rom Lemma 5.2, we recall the sp ectral representation T U 0 = X µ ∈ σ ( T U 0 ) µP µ , where P µ denotes the orthogonal pro jector on to the eigenspace Ker( T U 0 − µI ). As a consequence from standard spectral calculus, we ha ve Ω( a n ) = e − ia n T U 0 = X µ ∈ σ ( T U 0 ) e − ia n µ P µ . Since σ ( T U 0 ) is at most countable, there exist a subsequence denoted by ( a ′ n ) and some at most countable set { ω ( µ ) } µ ∈ σ ( T U 0 ) ⊂ S 1 suc h that e − ia ′ n µ → ω ( µ ) as n → ∞ for any µ ∈ σ ( T U 0 ) . Consequen tly , the unitary operators Ω( a ′ n ) = e − ia ′ n T U 0 con verge strongly to Ω ∞ := X µ ∈ σ ( T U 0 ) ω ( µ ) P µ , whic h is eviden tly a unitary op erator in L 2 + . □ HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 39 As b efore, let us denote by U ( t ) = Φ t ( U 0 ) = ( U ( t )Π U 0 ) + ( U ( t )Π U 0 ) ∗ − M ( U 0 ) the solution given b y the ﬂo w map for (HWM) as provided b y Theorem 2.1. W e deduce the following con vergence lemma. Lemma 10.2. L et ( t n ) b e a se quenc e in R such that Ω( t n ) → Ω ∞ str ongly with some unitary map Ω ∞ : L 2 + → L 2 + . Then U ( t n ) → U ∞ in H 1 / 2 , wher e the limit U ∞ b elongs to H 1 / 2 ( T ; Gr k ( C d )) and it satisﬁes E [ U ∞ ] = E [ U 0 ] , Π U ∞ ( z ) = M (( I − z Ω ∞ S ∗ ) − 1 Π U 0 ) for z ∈ D . Pr o of. W e adapt the arguments used in Section 9 to sho w this result. Step 1. Since Ω( t n ) → Ω ∞ strongly and by mirroring the proof of Lemmas 9.1 and 9.2, we deduce that U ( t n ) → U ∞ in L 2 and U ( t n ) ⇀ U ∞ in H 1 / 2 . Here the limit U ∞ b elongs to H 1 / 2 ( T ; Gr k ( C d )) and satisﬁes the explicit form ula Π U ∞ ( z ) = M  ( I − z Ω ∞ S ∗ ) − 1 Π U 0  for z ∈ D . F urthermore, from Lemma 8.1 w e obtain the in tertwining relation S ∗ U ∞ = U ∞ Ω ∞ S ∗ , (10.1) where U ∞ : L 2 + → L 2 + is the contraction giv en by ( U ∞ F )( z ) = M (( I − z Ω ∞ S ∗ ) − 1 F ) . for any F ∈ L 2 + and z ∈ D . Step 2. W e claim that U ∞ : L 2 + → L 2 + is unitary . In view of Theorem 8.1, this amounts to sho wing that Ker U ∞ = { 0 } holds. T o see this, let us deﬁne the T o eplitz op erator T U ∞ with symbol U ∞ ∈ H 1 / 2 ( T ; Gr k ( C d )). Since U ( t n ) → U ∞ w eakly and T U ( t n ) → T U ∞ strongly as operators, w e can pass to the limit n → ∞ in the Lax ev olution T U ( t n ) U ( t n ) = U ( t n ) T U 0 . This yields T U ∞ U ∞ = U ∞ T U 0 . (10.2) No w, b y exactly follo wing the arguments in the pro of of Lemma 9.3 and using the in tertwining relation (10.1) ab o ve, w e infer that T U 0 F = µ F and F  = 0 ⇒ T U ∞ U ∞ F = µ U ∞ F and U ∞ F  = 0 . Th us the map U ∞ : Ker( T U 0 − µI ) → Ker( T U ∞ − µI ) is injectiv e and, by a direct adaptation of the pro of of Lemma 9.4, we deduce that Ker U ∞ = { 0 } is trivial. This shows that U ∞ is unitary by Theorem 8.1. Step 3. Since U ∞ is unitary , w e see from (10.2) that T U ∞ = U ∞ T U 0 U ∗ ∞ . Therefore K U ∞ = U ∞ K U 0 U ∗ ∞ with the trace-class op erator K U ∞ = I − T 2 U ∞ and w e deduce that E [ U ∞ ] = T r( K U ∞ ) = T r( U ∞ K U 0 U ∗ ∞ ) = T r( K U 0 ) = E [ U 0 ] . In view of E [ U ( t n )] = E [ U 0 ] this implies that we must hav e strong conv ergence U ( t n ) → U ∞ in H 1 / 2 . □ 40 P A TRICK G ´ ERARD AND ENNO LENZMANN Pro of of Theorem 2.3. Let U 0 ∈ H 1 / 2 ( T ; Gr k ( C d )) b e an initial datum for (HWM). As usual, we denote b y U ( t ) = Φ t ( U 0 ) ∈ C ( T ; H 1 / 2 ) the solution pro vided b y Theorem 2.1. T o show that U ∈ AP( R ; H 1 / 2 ) holds, we use Bo c hner’s criterion for almost p eriodicity . Thus w e hav e to pro ve that, for every sequence ( a n ) in R , there exists a subsequence ( a ′ n ) such that the sequence of b ounded and contin uous functions R → H 1 / 2 ( T ; C d × d ) , t 7→ U ( t + a ′ n ) is uniformly conv ergent on R as n → ∞ . Indeed, giv en a sequence ( a n ) in R , w e c ho ose the subsequence ( a ′ n ) provided by Lemma 10.1 ensuring that Ω( a ′ n ) → Ω ∞ strongly as op erator with some unitary map Ω ∞ in L 2 + . F or t ∈ R , we deﬁne U ∞ ( t ) := Π U ∞ ( t ) + (Π U ∞ ( t )) ∗ − M (Π U 0 ) where we set Π U ∞ ( t, z ) := M  I − z Ω( t )Ω ∞ S ∗ ) − 1 Π U 0  with Ω( t ) = e − itT U 0 . Eviden tly , we ha ve that Ω( t )Ω( a ′ n ) → Ω( t )Ω ∞ strongly for an y t ∈ R . Thus by Lemma 10.2 applied to the sequence ( t n ) = ( t + a ′ n ), we deduce U ( t + a ′ n ) → U ∞ ( t ) in H 1 / 2 for any t ∈ R . No w, w e claim that w e hav e in fact uniform con v ergence on R , i.e., sup t ∈ R ∥ U ( t + a ′ n ) − U ∞ ( t ) ∥ H 1 / 2 → 0 as n → ∞ . (10.3) Note that the sup is ﬁnite, since U ∈ B C ( R ; H 1 / 2 ) and sup t ∈ R ∥ U ∞ ( t ) ∥ H 1 / 2 < ∞ thanks to E [ U ∞ ( t )] = E [ U 0 ] for all t ∈ R by Lemma 10.2. Th us we can ﬁnd a sequence ( t n ) in R such that sup t ∈ R ∥ U ( t + a ′ n ) − U ∞ ( t ) ∥ H 1 / 2 ≤ ∥ U ( t n + a ′ n ) − U ∞ ( t n ) ∥ H 1 / 2 + 2 − n . (10.4) Applying Lemma 10.1 again, w e can extract a subsequence ( t ′ n ) suc h that Ω( t ′ n ) → e Ω ∞ strongly with some unitary op erator e Ω ∞ . By the group property of { Ω( t ) } t ∈ R , w e immediately ﬁnd that Ω( t ′ n + a ′ n ) = Ω( t ′ n )Ω( a ′ n ) → e Ω ∞ Ω ∞ strongly . By Lemma 10.2, this implies that U ( t ′ n + a ′ n ) → U ∞ in H 1 / 2 , where the limit U ∞ satisﬁes Π U ∞ ( z ) = M (( I − z e Ω ∞ Ω ∞ S ∗ ) − 1 Π U 0 ) for z ∈ D . (10.5) On the other hand, from the explicit expression for U ∞ ( t ), we see that Π U ∞ ( t ′ n , z ) = M (( I − z Ω( t ′ n )Ω ∞ S ∗ ) − 1 Π U 0 ) for z ∈ D . Since Ω( t ′ n )Ω ∞ → e Ω ∞ Ω ∞ strongly , we also deduce from Lemma 10.2 that U ( t ′ n ) → U ∞ in H 1 / 2 , where the limit U ∞ is again given b y (10.5). Hence w e ha ve sho wn that ∥ U ( t ′ n + a ′ n ) − U ∞ ( t ′ n ) ∥ H 1 / 2 → 0 . HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 41 By passing to the subsequence ( t ′ n ) in (10.4) and taking the limit n → ∞ , we obtain the claimed uniform conv ergence (10.3). This sho ws that U ∈ AP( R ; H 1 / 2 ) holds. Finally , w e remark that the P oincar´ e recurrence as stated in Theorem 2.3 is an immediate consequence of Bohr’s c haracterization of almost p eriodicity . Moreov er, b y the standard fact that almost perio dic functions are uniformly contin uous, we deduce that { Φ t ( U 0 ) : t ∈ R } is a relatively compact subset in H 1 / 2 . The pro of of Theorem 2.3 is no w complete. □ App endix A. L WP in H s with s > 3 2 Lemma A.1 (L WP in H s with s > 3 2 ) . L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers and assume that s > 3 2 . Then, for every U 0 ∈ H s ( T ; Gr k ( C d )) , ther e exists a unique solution U ∈ C ( I ; H s ) of (HWM) with U (0) = U 0 , and I ⊂ R denotes its c orr esp onding maximal time interval of existenc e. Mor e over, we have glob al-in-time existenc e, i.e., I = R , pr ovide d that sup t ∈ I ∥ U ( t ) ∥ H s < ∞ . Pr o of. The pro of of Lemma A.1 follo ws from a direct adaptation of [ 13 ][App endix D], where the analogous lo cal well-posedness result for (HWM) p osed on R for H s -data with s > 3 2 is prov ed by a Kato-type iteration sc heme. Note that the com- m utator estimates and fractional Leibniz rule used in [ 13 ] can b e directly adapted to the setting on the torus; see, e.g., [ 2 ] for a systematic treatment of fractional Leibniz rule estimates on T from the corresp onding estimates for functions on R . □ R emark. A closer inspection of the proof shows that an a-priori Lipschitz b ound sup t ∈ I ∥ ∂ θ U ( t ) ∥ L ∞ < ∞ w ould be suﬃcient to deduce global existence in Lemma A.1 abov e. Ho wev er, as already pointed out in the introduction in Section 1, the Lax structure for (HWM) fails to provide suc h an a-priori bound. F urthermore, by adapting the discussion in [ 13 ] for (HWM) on R , we deduce the follo wing res ult for the op erator-v alued evolution equation d dt U ( t ) = B U ( t ) U ( t ) for t ∈ I , U (0) = I (A.1) where I ⊂ R is a time in terv al with 0 ∈ I and giv en U ∈ C ( I ; H s ( T ; Gr k ( C d ))) with some s > 3 2 together with the unbounded and (formally) sk ew-adjoint operator B U = i 2 ( D T U + T U D ) − i 2 T | D | U acting on L 2 + = L 2 + ( T ; C d × d ) with D = − i∂ θ . Lemma A.2. L et d ≥ 2 and 1 ≤ k ≤ d − 1 b e inte gers and supp ose that U ∈ C ( I ; H s ( T ; Gr k ( C d ))) solves (HWM) with some s > 3 2 on the time interval I ⊂ R with 0 ∈ I . Then ther e exists a unique solution U : I → L ( L 2 + ) of (A.1) with the fol lowing pr op erties. (i) The map I 7→ L 2 + with t 7→ U ( t ) F is c ontinuous for every F ∈ L 2 + . (ii) The e quation ∂ t U ( t ) = B U ( t ) U ( t ) holds in H − 1 + ( T ; C d × d ) for any t ∈ I . (iii) U ( t ) : L 2 + → L 2 + is a unitary map for any t ∈ I . 42 P A TRICK G ´ ERARD AND ENNO LENZMANN (iv) U ( t ) F ∈ H 1 + for any F ∈ H 1 + = H 1 ∩ L 2 + . (v) The op er ator B U ( t ) : H 1 + ⊂ L 2 + → L 2 + is essential ly skew-adjoint. R emark. Note that (iii) means that there exists a unique skew-adjoin t extension B ∗ U ( t ) = − B U ( t ) with some domain Dom( B U ( t ) ) and H 1 + is dense in Dom( B U ( t ) ) with resp ect to the graph norm of B U ( t ). Pr o of. The pro of follows from a direct adaptation from the arguments in the pro of of [ 13 ][Lemma D.2]. See also the remark follo wing [ 13 ][Lemma D.2] ab out property (iii). □ App endix B. Half-harmonic maps and solitary w a ves Let d ≥ 2 and 1 ≤ k ≤ d − 1 b e integers. W e recall that critical p oin ts Q ∈ H 1 / 2 ( T ; Gr k ( C d )) of the energy functional E [ U ] in (1.1) are half-harmonic maps and they are (weak) solutions of [ Q , | D | Q ] = 0 . (B.1) Eviden tly , the half-harmonic maps corresp ond to stationary solutions of (HWM) with ﬁnite energy . As observed in [ 17 ], we hav e indeed tr aveling solitary waves for (HWM) with given velocity v ∈ ( − 1 , 1). The corresp onding proﬁles Q v ∈ H 1 / 2 ( T ; Gr k ( C d )) are (weak) solutions of (B.1) in the generalized form [ Q , | D | Q ] − i 2 v ∂ θ Q = 0 . (B.2) Sp e cial c ase Gr 1 ( C 2 ) ∼ = S 2 . In this case all half-harmonic maps are explicitly kno wn in closed form. Adapting this result to our matrix-v alued formulation, we ﬁnd that all half-harmonic maps U ∈ H 1 / 2 ( T ; Gr 1 ( C 2 )) are of the form Q ( θ ) = U 0 B (e iθ ) B (e iθ ) 0 ! U ∗ (B.3) where U ∈ C 2 × 2 is any constan t unitary matrix, and B : D → D with B ( z ) = e iφ m Y k =1 B ( z , a k ) is an y ﬁnite Blaschke pr o duct of degree m ∈ N with zeros a 1 , . . . , a m ∈ D and the Blasc hke factors B ( z , a ) = z − a 1 − az for a  = 0 and B ( z , 0) = z . F rom (B.3) a direct calculation shows that ener gy quantization of half-harmonic maps with E [ Q ] = π · m where m is the degree of the Blaschk e pro duct in Q . The case m = 0 yields the trivial case of constan t half-harmonic maps, whereas m = 1 corresp onds to ground state half-harmonic maps (i.e. nontrivial half-harmonic maps that globally minimize the energy). Moreo ver, by translating the results in [ 17 ] into matrix- v alued notation, we recall that the tra veling solitary proﬁles Q v with v ∈ ( − 1 , 1) can b e written as Q v ( θ ) = U v (1 − v 2 ) 1 / 2 B (e iθ ) (1 − v 2 ) 1 / 2 B (e iθ ) − v ! U ∗ , (B.4) with the energy E [ Q v ] = (1 − v 2 ) · π · m . Note that E [ Q v ] → 0 as | v | → 1, and hence we can construct tra veling solitary wa ves with arbitrarily small energy . HWM ON T : GWP IN H 1 / 2 AND ALMOST PERIODICITY 43 Gener al c ase Gr k ( C d ) . F or the general Grassmannians Gr k ( C d ) as targets, we can easily build half-harmonic maps as follo ws. Let d ≥ 2 and 1 ≤ k ≤ d − 1 b e in tegers, where without loss of generalit y w e can assume that d ≥ 3 holds, since the remain- ing case Gr 1 ( C 2 ) is already understo od (see ab o ve). T o construct half-harmonic maps from T into Gr k ( C d ), w e use that elements in Gr 1 ( C 2 ) can b e naturally em- b edded in to Gr k ( C d ) as follo ws. Let Q ∈ H 1 / 2 ( T ; Gr 1 ( C 2 )) be a half-harmonic map, cf. (B.3), and deﬁne e Q ( θ ) =  Q ( θ ) 0 0 J  , where J is a constant matrix in Gr k − 2 ( C d − 2 ), e.g., take J =  1 p 0 0 − 1 q  with p, q ∈ { 0 , . . . , d − 2 } suc h that T r( J ) = p − q = d − 2 k . W e readily chec k that e Q ∈ H 1 / 2 ( T ; Gr k ( C d )) solv es (B.3) and th us it is a half-harmonic map. Likewise, w e can easily generalize this construction to obtain trav eling solitary wa ves for (HWM) with any target Gr k ( C d ) by using the kno wn case Gr 1 ( C 2 ). 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The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

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