Borel Spectrum of Operators on Banach Spaces

BOREL SPECTR UM OF OPERA TORS ON BANA CH SP A CE S MOHAMMED Y AHDI Abstract. The p ap e r inv estigates t he variation of the sp e ctrum of op e r ators in infinite dimensiona l Banach sp ac es. In p arti c ular, it is sh own that the sp e ctrum function is Bor el fr om the sp ac e of b ounde d op er ators on a sep ar able Banach sp ac e; e quipp e d with the str ong op er ator top olo g y, into the Polish sp ac e of co mp act subsets of the close d unit disc of the c omplex plane; equip p e d with the Hausdorff top olo gy. R emarks and r esults ar e g i ven when other top olo gies ar e use d. 1. Preliminar y Let X b e a n infinite dimensio nal Banach space. W e denote by T an arbitra r y bo unded op erato r on X and by I the ident ity oper ator on X . Le t D b e the closed unit disc of the complex plane C . The restr iction on D of the sp ectrum of a n op erator T , denoted by σ ( T ), is defined a s follows: σ ( T ) := { λ ∈ D ; ( T − λI ) is not inv ertible } . Recently , essential sp ectra of some matr ix op era tors on Banach space s (see [3]) a nd sp ectra of some blo ck o per ator ma trices (see [5]) were in vestigated, with applica - tions ro differential and transp or t op erators This pap er inv estiga tes the v ariatio ns of the sp ectrum σ ( T ) as T v ar ies over the s pace L ( X ) of all bo unded o per ator on the Bana ch space X . First, we intro duce the sets and the top ologies re quired for this study . W e deno te b y • K ( D ) the set of all co mpact subsets of the close d unit disc D of the complex plane C , • σ the sp ectrum function defined from L ( X ) into K ( D ) that maps an op er- ator T to its s p ectr um σ ( T ). The set K ( D ) is endow ed with the Hausdor ff top olo gy generated b y the fa milies of all subsets in one of the following fo rms  F ∈ K ( D ); F ∩ V 6 = ∅  and  F ∈ K ( D ); F ⊆ V  , 2000 Mathematics Subje c t Classific ation. Primary 54C10, 47A10, 54H05. Key wor ds and phr ases. Bor el F unction, Banach Space, Op erator, Po lish Space, Spectrum. 1 2 MOHAMMED Y AHDI for V an op en subset of D . Therefore, K ( D ) is a Polish space, i.e., a se pa rable metrizable co mplete space, sinc e D is Polish (see [7],[8] or [2]). It is shown b elow that we can reduce the families that genera te the ab ov e Hausdor ff topo logy . Prop ositio n 1.1. L et K ( D ) b e the set of c omp act su bsets of the close d unit disc D . Then K ( D ) e quipp e d with t he Hausdorff top olo gy is a Polish sp ac e; wher e the Bor el structur e is gener ate d by one of the fol lowing two families n { K ∈ K ( D ) : K ∩ V 6 = ∅ } ; V o p en in D o n { K ∈ K ( D ) : K ⊂ V } ; V op en in D o . Pr o of. Let V b e a n op en subset of D . There exis ts a decrea sing se quence of op en subsets ( O n ) n ∈ N such th at V c = T n ∈ N O n ; for example O n = { x ∈ D : di s t ( x, V c ) ≤ 1 n } . W e hav e { K ∈ K ( D ) : K ∩ V 6 = ∅} c = { K ∈ K ( D ) : K ⊆ V c } = \ n ∈ N { K ∈ K ( D ) : K ⊆ O n } . On the other hand, { K ∈ K ( D ) : K ⊆ V } c = { K ∈ K ( D ) : K ∩ V c 6 = ∅ } = { K ∈ K ( D ) : K ∩ O n 6 = ∅ , ∀ n ∈ N } = \ n ∈ N { K ∈ K ( D ) : K ∩ O n 6 = ∅} . Indeed, if for all n ∈ N , there exists x n ∈ K ∩ O n , then there ex ists a subs equence ( x n k ) k of ( x n ) n that conv erges to x ∈ K , and x ∈ T n O n since ( O n ) n is decre a sing.  2. Norm Opera tor Topolo gy and the Spectrum Function W e equip L ( X ) with the canonical no rm of operator s defined by k T k = sup x ∈ B X k T ( x ) k , where B X is the unit ball of X . Note that the map σ : T 7− → σ ( T ) is not contin uous when L ( X ) is endow ed with its canonical norm. Indeed, the op er a tors T n = (1 + 1 n ) I conv erg e to the ident ity I while σ ( T n ) = ∅ and σ ( I ) = { 1 } . How ever, w e hav e the following result. Prop ositio n 2.1. L et X b e a Banach sp ac e, ( L ( X ) , k . k ) t he sp ac e of b ounde d op er ators e quipp e d with the norm of op er ators, and K ( D ) t he set of c omp act subsets of the unit disc D e quipp e d with the Hausdorff top olo gy. Then the sp e ctrum map σ :  L ( X ) , k . k  − → K ( D ) T 7− → σ ( T ) BOREL SPECTRUM OF OPERA TORS ON BANA CH SP ACES 3 is upp er-semi c ontinuous . Pr o of. Let V b e an op en subset D . By pr op osition 1.1, it is only need to show that the s et O V = { T ∈ L ( X ); σ ( T ) ⊆ V } is k . k - o p e n in L ( X ). Let T 0 be fixed in O V . Since σ ( T 0 ) ∩ D ⊆ V , then for all λ ∈ D \ V , the op erator ( T 0 − λI ) is inv er tible and the map λ ∈ D \ V 7→ ( T 0 − λI ) − 1 is contin uous (see [ ? ]). It follows tha t sup λ ∈ D \ V   ( T − λI ) − 1   < + ∞ since D \ V is compa ct. P ut δ = inf λ ∈ D \ V 1 k ( T 0 − λI ) − 1 k > 0 . Let T ∈ L ( X ) such tha t k T − T 0 k < δ . F or any λ ∈ D \ V we hav e k ( T − λI ) − ( T 0 − λI ) k = k T − T 0 k < 1 k ( T 0 − λI ) − 1 k · Thu s, ( T − λI ) is in vertible and hence λ / ∈ σ ( T ). In other terms , σ ( T ) ⊆ V for a ll T ∈ L ( X ) with k T − T 0 k < δ . Therefor e O V is an o pen s ubset of  L ( X ) , k . k  .  3. Strong Opera tor Topology and the Spectr u m Function Consider now L ( X ) equipp ed with the str ong op erator top ology S op ( see [4]). In general, L ( X ) eq uipp ed with the str ong op erator topo logy is no t a po lish space (since it is not a Baire space). How ever, if X is separa ble, then ( L ( x ) , S op ) is a standard Borel space. Indeed, it is Bore l- isomorph to a Borel subset of the Polish space X N equipp e d with the norm pro duct top olog y via the ma p ϕ :  L ( X ) , S op  − →  X N , P  T 7− → ( T z n ) n ∈ N , where { z n , n ∈ N } is a dense Q -vector spa ce in X . Let us c heck how this top ology on L ( x ) affects the s pectr um function. Theorem 3 .1. F or any sep ar able infin ite dimensional Banach X , the map σ : L ( X ) − → K ( D ) T 7− → σ ( T ) , which maps a b ounde d op er ator t o its sp e ct rum, is Bor el when L ( X ) is endowe d with the str ong op er ator t op olo gy S op and K ( D ) with t he Hausddorf top olo gy. Pr o of. As K ( D ) is equipp ed with the Hausdo rff top olog y , it follows from the prop o- sition 1.1, tha t it is enough to show that for any op en subset V o f the disc D , the subset E V = { T ∈ L ( X ) : σ ( T ) ∩ V 6 = ∅ } 4 MOHAMMED Y AHDI is Borel in  L ( X ) , S op  . Let V b e a fixe d o pe n subset o f D . W e hav e E V = P L ( X ) (Ω) , where P L ( X ) stands for the canonical pro jection of L ( X ) × D onto L ( X ), and Ω =  ( T , λ ) ∈ L ( X ) × V : λ ∈ σ ( T )  . By a descriptive set theory result fro m [9], to show that E V is a Borel s e t it suffices to show that Ω is a Borel set with K σ vertical sections. F or T ∈ L ( X ), the vertical section of the set Ω ⊆ L ( X ) × D along the direction T is given by Ω( T ) =  λ ∈ D : ( T , λ ) ∈ Ω  =  λ ∈ D : λ ∈ V ∩ σ ( T )  = σ ( T ) ∩ V . Thu s, it is a K σ of D . Now, we need to pr ov e that Ω is a Bor el s e t. Put ∆ =  ( T , λ ) ∈ L ( X ) × D : λ ∈ σ ( T )  . Therefore Ω = ∆ ∩ L ( X ) × V , Hence, to finish the pro of, it is enough to prov e the following claim. Claim: ∆ is a Bo rel set of L ( X ) × D . First, note tha t ∆ = A ∪ B with • A =  ( T , λ ) ∈ L ( X ) × D : T − λI is no t isomorph to its range  • B =  ( T , λ ) ∈ L ( X ) × D : ( T − λI )( X ) is not dense in X  . Indeed, if T − λI is an isomorphism o n to its r ange, then ( T − λI )( X ) is a closed subspace that will be strict if λ ∈ σ ( T ), and thus not dense in X . On the o ther hand, since X is separ able, there exists a countable and dense subset Y in the spher e S X of X , and there exists a dense sequence { x n } n ∈ N in X . Now, we will show that A a nd B are Bo rel sets. Let ( T , λ ) ∈ L ( X ) × D . F rom the definition o f A , W e have ( T , λ ) ∈ A if a nd only if ∃ ( z n ) n ∈ N ⊆ S X : lim n →∞ k ( T − λI ) z n k = 0 . In other term, this is equiv alent to ∃ ( z n ) n ∈ N ⊆ Y , ∀ k ≥ 1 ∃ N k ∈ N ∀ n ≥ N k : k ( T − λI ) z n k < 1 k . BOREL SPECTRUM OF OPERA TORS ON BANA CH SP ACES 5 By choo sing the subsequenc e ( z N k ) k ∈ N instead of ( z n ) n ∈ N , the previous statement is equiv alent to ∃ ( z n ) n ∈ N ⊆ Y , ∀ k ≥ 1 ∃ N k ∈ N : k ( T − λI ) z N k k < 1 k , or again, ∀ k ≥ 1 , ∃ x ∈ Y : k T x − λx k < 1 k . Therefore, A = \ k ≥ 1 [ x ∈Y A x k with A x k =  ( T , λ ) ∈ L ( X ) × D : k T x − λx k < 1 k  . Since L ( X ) is equipp ed with the the s trong o p e r ator conv erge nce S op , it follows that A x k are op en sets. Hence, A is a Borel set. On the other hand, “( T − λI )( X ) is not dense in X ” is eq uiv a lent to ∃ y ∈ S X and ∃ k ≥ 1 such that ∀ x ∈ X : k y − ( T − λI ) x k ≥ 1 k , or again, ∃ y ∈ Y and ∃ k ≥ 1 s uch that ∀ n ∈ N : k y − ( T − λI ) x n k ≥ 1 k . Therefore B = [ y ∈Y [ k ∈ N \ n ∈ N B y k,n with B y k,n = n ( T , λ ) ∈ L ( X ) × D ; k y − ( T − λI ) x n k ≥ 1 k o . Similarly to A x k , it is not difficult to see that the sets B y k,n are Borel sets. Hence B is also a Borel set. This proves the claim and ends the pro of of the theor em 3.1.  References [1] Aup etit, R. , A primer on sp ectr al the ory , Springer-V erl ag, New y ork, 1991. [2] Christensen, J.P .R., T op ology and Bor el structure, North-H ol land Math. St ud., 10 , 1974. [3] Damak, M. and Jeribi , A., O n The Es s en tial Sp ectra Of Matrix Op erators A nd Applications, Ele ctr onic Journal of Differ ential Equations, V ol. 2007 No. 11 : 1–16. [4] Dunford, N. and Sch wartz , J., Linear operator, Part. I , DA Wiley-Interscience Public ation New Y ork-London-Sydney, 1971. [5] Jeribia, A., M oall a, N . and W alhaa, I., Spectra of some blo ck op erator matrices and applica- tion to transp ort operators, Dokl. Akad . Nauk SSSR (N.S.) V olume 351 Issue 1 : 315–325, March 2009. [6] Kech ris, A.S., Classical Descri ptiv e Set Theory , Springer- V erlag New Y or k, 1995. [7] Kech ris, A.S. and Louv eau, A. Descriptive set theory and the structure of sets of uniqueness, L ondon Math. So c., L e c tur e Notes series 12 8 , 1987. [8] Kuratowski, K., T op ology , V ol. I I, A c ademic Pr e ss New Y ork, 1966. [9] Saint Ra ymond, J., Bor´ eliens ` a coupes K σ , Bul l. So c. math. F r anc e 104 : 389–400, 1976. 6 MOHAMMED Y AHDI Dep ar tment of M a thema tics and Computer Science, Ursinus College, Collegev ille, P A 19 426, USA E-mail addr ess : myahdi@ursin us.edu