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APPEARED IN BULLETIN OF THE

arXiv:math/9302216v1 [math.FA] 24 Feb 1993RESEARCH ANNOUNCEMENTAPPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 31, Number 1, July 1994, Pages 44-49LYAPUNOV THEOREMS FOR BANACH SPACESY. Latushkin and S. Montgomery-SmithAbstract.

We present a spectral mapping theorem for semigroups on any Banachspace E. From this, we obtain a characterization of exponential dichotomy for nonau-tonomous differential equations for E-valued functions. This characterization is givenin terms of the spectrum of the generator of the semigroup of evolutionary operators.1.

IntroductionLet us consider an autonomous differential equation y′ = Ay in a Banach spaceE, where A is a generator of continuous semigroup {etA}t≥0; that is, the solutionof the differential equation satisfies y(t) = etAy(0), t ≥0. A classical result of A.M. Lyapunov (see, e.g., [DK]) shows that for bounded A, the spectrum σ(A) of Ais responsible for the asymptotic behavior of y(t).

For example, if Re σ(A) < 0,then the trivial solution is uniformly asymptotically stable, that is, ∥etA∥→0 ast →∞. This fact follows from the spectral mapping theorem (see, e.g., [N, p. 82]):(1)σ(etA)\{0} = exp(tσ(A)),t ̸= 0,which always holds for bounded A.For unbounded A, equation (1) is not always true.

Moreover, there are examplesof generators A (see [N, p. 61]) such that even Re(σ(A)) ≤s0 < 0 does not guaranteeσ(eA) ⊂D = {|z| < 1} or ∥etA∥→0 as t →∞. Since σ(A) does not characterizethe asymptotic behavior of the solutions y(t), we would like to find some othercharacterization that still does not involve solving the differential equation (i.e.,finding σ(etA)).1991 Mathematics Subject Classification.

Primary 4706, 47B38; Secondary 34D20, 34G10.Key words and phrases. Hyperbolicity, evolution family, exponential dichotomy, weighted com-position operators, spectral mapping theorem.Received by the editors February 19, 1993c⃝1994 American Mathematical Society0273-0979/94 $1.00 + $.25 per page1

2Y. LATUSHKIN AND S. MONTGOMERY-SMITHIn this article we solve precisely this problem in the following manner.

Considerthe space Lp(R; E) of E-valued functions for 1 ≤p < ∞and the semigroup {etB}t≥0of evolutionary operations (also called weighted translation operators)(2)(etBf )(x) = etAf(x −t),t ≥0,generated by the operator B = −ddx + A, x ∈R. It turns out that it is σ(B) inLp(R; E) that is responsible for the asymptotic behavior of y(t) in E. For example,Re(σ(B)) < 0 on Lp(R; E) implies that ∥etA∥→0 as t →∞on E.We will also consider the well-posed nonautonomous equation y′ = A(t)y(t).Instead of the semigroup given by (2), we consider in Lp(R; E) the semigroup(3)(etDf )(x) = U(x, x −t)f(x −t),x ∈R, t ≥0 .Here U(t, s), t ≥s, is the evolutionary family (propagator) for the nonautonomousequation.

We will show that σ(D) characterizes the asymptotic behavior of y(t).The order of proofs will be as follows. We will first show the spectral mappingtheorem for the autonomous case (2).

We will also consider a similar theorem forthe evolutionary semigroup in the space of the periodic functions. This theoremwill give us a variant of Greiner’s spectral mapping theorem (see [N, p. 94]) forany C0-semigroup {etA} in a Banach space.

This variant also is a direct general-ization of Gerhard’s spectral mapping theorem in Hilbert space for generators withresolvent bounded along iR (see [N, p. 95]). Then we will obtain the spectral map-ping theorem for the nonautonomous case (3) using a simple change of variablesargument to reduce it to the autonomous case (2).We will be considering not only stability but also the exponential dichotomy (hy-perbolicity) for the solutions of the equation y′ = A(t)y(t) in E. In the theory ofdifferential equations with bounded coefficients, exponential dichotomy is an impor-tant tool used, for example, in proving instability theorems for nonlinear equationsand determining existence and uniqueness of bounded solutions and Green’s func-tions (see, e.g., [DK]).

The spectral mapping theorem given here for the semigroup(3) allows one to extend these ideas to the case of unbounded coefficients.In turns out that the condition 0 /∈σ(D)—or equivalently σ(etD)∩T = ∅, t > 0,T = {|z| = 1} on Lp(R; E)—is equivalent to the hyperbolic behavior of a specialkind for the solutions. We will call this spectral hyperbolicity.Note that if theU(t, s) are invertible for (t, s) ∈R2, that is, (3) is extendible to a group, then thespectral hyperbolicity is the same as the exponential dichotomy (or hyperbolicity)in the usual (see, e.g., [DK]) sense.

Therefore, the spectrum σ(etD) for nonperiodicA(·) plays the same role in the description of exponential dichotomy as the spectrumof the monodromy operator does in the usual Floquet theory for the periodic case.However, ordinary hyperbolicity is not equivalent (see [R]) to spectral hyperbolicityin the semigroup case and thus cannot be characterized in terms of σ(etD) only.For a Hilbert space, we were able to characterize hyperbolicity in terms of otherspectral properties of etD.Finally, the results of this article can be generalized to the case of the variationalequation y′(t) = A(ϕtx)y(t) for a flow {ϕt} on a compact metric space X or fora linear skew-product flow ˆϕt : X × E →X × E : (x, y) 7→(ϕtx, Φ(x, t)y), t ≥0(see [CS, H, LS, SS] and references contained therein). Here Φ: X × R+ →L(E)is a cocycle over ϕt, that is, Φ(x, t + s) = Φ(ϕtx, s)Φ(x, t), Φ(x, 0) = I.

Let us

LYAPUNOV THEOREMS FOR BANACH SPACES3recall (see [SS]) that one of the purposes of the theory of linear skew-product flowswas to aid in studying the equation y′ = A(t)y for the case of almost periodicA(·). To answer the question when ˆϕt is hyperbolic (or Anosov), instead of (3) oneconsiders the semigroup of so-called weighted composition operators (see [CS, J,LS]) on Lp(X; µ; E):(4)(T tf )(x) =dµ ◦ϕ−tdµ1/pΦ(ϕ−tx, t)f(ϕ−tx),x ∈X, t ≥0 .Here µ is a ϕt-quasi-invariant Borel measure on X.

As above, the condition σ(T t)∩T = ∅is equivalent to the spectral hyperbolicity of the linear skew-product flow ˆϕt.The spectral hyperbolicity coincides with the usual hyperbolicity if Φ(x, t), x ∈X,t ≥0, are invertible or compact operators. Unlike the finite-dimensional case (see[M]), the hyperbolicity of ˆϕt does not generally imply the condition σ(T t)∩T = ∅.For Hilbert space the condition of hyperbolicity of ˆϕt can also be described in termsof other spectral properties of T t. We will not include these generalizations in thispaper.We point out that the investigation of evolutionary operators (2) and (3) has along history [Ho].

More recently, significant progress has been made in [BG, N, P,R] (see [R] for detailed bibliography), and this list does not pretend to be complete.A detailed investigation of weighted composition operators (4) for Hilbert spacesE and connections with the spectral theory of linear skew-product flows [SS] andother questions of dynamical systems theory and a bibliography may be found in[LS].Notation. L(E) (correspondingly Ls(E)) denotes the set of bounded operators onE with the uniform (correspondingly strong) topology; ρ(A) denotes the resolventset of the operator A; | denotes the restriction of an operator; Cb(R; E) denotes thespace of continuous bounded E-valued functions on R with the supremum norm,and C0b (R; E) denotes the subspace of functions vanishing at infinity.2.

Autonomous caseLet A be a generator of any C0-semigroup in a Banach space E. Consider theC0-semigroup (2) on the space Lp([0, 2π); E), 1 ≤p < ∞, that is, (etBf )(x) =etAf((x −t)(mod 2π)).Theorem 1. The following are equivalent:(1) 1 ∈ρ(e2πA) on E;(2) 0 ∈ρ(B) on Lp([0, 2π); E);(3) 1 ∈ρ(e2πB) on Lp([0, 2π); E).In the main part of the proof (2)⇒(1), we modify the idea of [CS].

Let us assumethat (2) is fulfilled, but for each 0 < ε < 12, there is a y ∈E such that ∥e2πAy−y∥< εand ∥y∥= 1 (and hence ∥e2πAy∥≥12).Let ρ(x) =32πx −1 for x ∈[ 2π3 , 4π3 ),ρ(x) = 0 for x ∈[0, 2π3 ), and ρ(x) = 1 for x ∈[ 4π3 , 2π).Define the functionf ∈Lp([0, 2π); E) by f(x) = (1 −ρ(x))e(2π+x)Ay + ρ(x)exAy, x ∈[0, 2π). Then forc = max{∥exA∥: x ∈[0, 2π)}, one has ∥e2πAy∥= ∥e(2π−x)AexAy∥≤c∥exAy∥.

But,in contradiction with (2),∥f ∥pp ≥Z 2π4π/3∥exAy∥p dx ≥2π3 c−p∥e2πAy∥p ≥2π3 c−p2−p

4Y. LATUSHKIN AND S. MONTGOMERY-SMITHand∥Bf ∥p ≤2π3 cε .Theorem 1 implies the following variant of Greiner’s spectral mapping theorem(see [N, p. 94]) for a C0-semigroup {etA} in Banach space E.Theorem 2.

1 ∈ρ(e2πA) if and only if (a) iZ ⊂ρ(A) and (b) there is a constantC such that for any finite sequence {yk} ⊂EXk(A −ik)−1yke−ikxLp([0,2π);E)≤CXkyke−ikxLp([0,2π);E).Obviously, if E is a Hilbert space and p = 2, then Parseval’s identity allows oneto replace (b) by the condition sup{∥(A −ik)−1∥: k ∈Z} < ∞. This gives thefamous spectral mapping theorem of Gerhard [N, p. 95].Let us consider (2) on Lp(R; E), 1 ≤p < ∞.

The spectrum σ(B) now is invariantunder the translations along the imaginary axis. Moreover, we have the followingresult.Theorem 3.

For each t > 0 the following are equivalent:(1) σ(etA) ∩T = ∅on E;(2) 0 ∈ρ(B) on Lp(R; E);(3) σ(etB) ∩T = ∅on Lp(R; E).Thus, the spectral mapping theorem is valid for (2). If {etA}t∈R is a group, then(1) is the same as the exponential dichotomy of the autonomous equation y′ = Ayon R.3.

Nonautonomous caseConsider the well-posed nonautonomous equation y′(t) = A(t)y(t). By “well-posed” we mean that we assume the existence of a jointly strongly continuousevolutionary family U(t, s) ∈Ls(E), t ≥s, with the properties U(t, t) = I, U(t, r) =U(t, s)U(s, r), and ∥U(r, s)∥≤Ceβ(t−s), t ≥r ≥s.

In fact, U is a propagator forthe equation y′(t) = A(t)y(t), that is, y(t) = U(t, s)y(s). The spectral mappingtheorem is valid for (3).Theorem 4.

Let (3) be a C0-semigroup on Lp(R; E), 1 ≤p < ∞. Then σ(D) isinvariant under translations along the imaginary axis and the following are equiv-alent:(1) 0 ∈ρ(D) on Lp(R; E);(2) σ(etD) ∩T = ∅on Lp(R; E), t > 0.To outline the proof of (1)⇒(2), let us consider the semigroup (etBh)(s, x) =U(x, x −t)h(s −t, x −t), (s, x) ∈R2, t > 0, on the space Lp(R × R; E) =Lp(R; Lp(R; E)) and perform a change of variables u = s + x, v = x.

More pre-cisely, consider the isometry J on Lp(R × R; E) defined by (Jh)(s, x) = h(s + x, x).Then one has that JetB = (I ⊗etD)J and JB = (I ⊗D)J, where A1 ⊗A2means that A1 acts on h(·, x) and A2 acts on h(s, ·). Since etB can be writtenas (etBf )(s) = etDf(s −t) for f : R →Lp(R; E): s 7→h(s, ·), one can apply to etBthe part (2)⇒(3) of Theorem 3.

LYAPUNOV THEOREMS FOR BANACH SPACES5Definition. The evolutionary family {U(x, s)}x≥s is called hyperbolic if there ex-ists a projection-valued function P : R →L(E), P ∈Cb(R; Ls(E)), and M, λ > 0such that for all x ≥s:(1) P(x)U(x, s) = U(x, s)P(s); and(2) ∥U(x, s)y∥≤Me−λ(x−s)∥y∥if y ∈Im P(s);∥U(x, s)y∥≥M −1eλ(x−s)∥y∥if y ∈Ker P(s).The evolutionary family {U(x, s)}x≥s is called spectrally hyperbolic if, in addition:(3) Im U(x, s)| Ker P(s) is dense in Ker P(x).Note that the second inequality in (2) implies only left invertibility of the re-striction U(x, s)| Ker P(s), while (3) guarantees its invertibility.

If the evolutionaryfamily {U(x, s)}(x,s)∈R2 consists of invertible operators, then spectral hyperbolicityis equivalent to the hyperbolicity and is the same as exponential dichotomy [DK] ofthe equation y′ = A(t)y on R. If dim Ker P(s) < d < ∞, then obviously (2) alwaysimplies (3). This also happens if the U(x, s) are compact operators in E [R1].Theorem 5.

The evolutionary family {U(x, s)} on the separable Banach space Eis spectrally hyperbolic if and only if 0 ∈ρ(D) on Lp(R; E).Remark. The space Lp may be replaced by the space C([0, 2π); E) in Theorem 1and by C0b (R; E) in Theorems 2 to 5.

The separability assumption in Theorem 5was recently removed [LR].A remarkable observation in [R] shows that the hyperbolicity of {U(x, s)}x≥s(unlike the spectral hyperbolicity) does not generally imply (2) in Theorem 4 forinfinite-dimensional E. However, we are able to give the following characterizationof hyperbolicity under the assumptions that E is a Hilbert space and that for somer > 0 the function x 7→U(x + r, x) is a continuous function from R to L(E).Theorem 6. The evolutionary family {U(x, s)}x≥s is hyperbolic on a separableHilbert space E if and only if there exists a projection P on L2(R; E) such that forsome t > 0:(1) etDP = PetD;(2) σ(etD| Im P) ⊂D;(3) etD| Ker P is left invertible and σ((etD| Ker P)†) ⊂D;(4) Ker P ⊖Tn≥0 Im(entD| Ker P) is invariant with respect to multiplicationsby the functions from Cb(R; R).Each projection P with these properties has a form (Pf)(x) = P(x)f(x) for aprojection-valued function P ∈Cb(R; L(E)).For the left-invertible operator T the notation T † stands for its Moore-Penroseleft inverse: T †u = v if u = T v, and T †u = 0 for u⊥Im T .

Note that (1), (2), (3)imply the left invertibility of zI−etD for all z ∈T and the formula P =1(2πi)RT(zI−etD)† dz which gives the Riesz projection on σ(etD) ∩D if σ(etD) ∩T = ∅.References[BG] A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinarydifferential operators, Oper. Theory Adv.

Appl., vol. 56, Birkha¨user, Basel, 1992, pp.

90–119.[CS]C. Chicone and R. Swanson, Spectral theory of linearizations of dynamical systems, J.Differential Equations 40 (1981), 155–167.

6Y. LATUSHKIN AND S. MONTGOMERY-SMITH[DK] J. Daleckij and M. Krein, Stability of differential equations in Banach space, Transl.

Math.Mono., vol. 43, Amer.

Math. Soc., Providence, RI, 1974.[H]J.

Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monographs, vol.

25,Amer. Math.

Soc., Providence, RI, 1988.[Ho]J. S. Howland, Stationary scattering theory for time-dependent hamiltonians, Math.

Annal.207 (1974), 315–335.[J]R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980), 366–387.

[LM] Y. Latushkin and S. Montogomery-Smith, Evolutionary semigroups and Lyapunov theoremsin Banach spaces, J. Funct. Anal.

(to appear). [LR] Y. Latushkin and T. Randolph, Dichotomy of differential equations on Banach spaces andan algebra of weighted translation operators, Trans.

Amer. Math.

Soc., submitted.[LS]Y. Latushkin and A. Stepin, Weighted translations operators and linear extensions of dy-namical systems, Russian Math.

Surveys 46 (1991), 95–165.[N]R. Nagel (ed.

), One parameters semigroups of positive operators, Lecture Notes in Math.,vol. 1184, Springer-Verlag, Berlin, 1984.[M]J.

Mather, Characterization of Anosov diffeomorphisms, Indag. Math.

30 (1968), 479–483.[P]K. Palmer, Exponential dichotomy and Fredholm operators, Proc.

Amer. Math.

Soc. 104(1988), 149–156.[R]R.

Rau, Hyperbolic evolution groups and exponentially dichotomic evolution families, J.Funct. Anal.

(to appear). [R1], Hyperbolic evolutionary semigroups on vector-valued function spaces, SemigroupForum 48 (1994), 107–118.[SS]R.

Sacker and G. Sell, Dichotomies for linear evolutionary equations in Banach spaces, IMApreprint no. 838, 1991.Department of Mathematics, University of Missouri, Columbia, Missouri 65211E-mail address, Y. Latushkin: mathyl@mizzou1.missouri.eduE-mail address, S. Montgomery-Smith: stephen@mont.cs.missouri.edu

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