On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces

Contents Int ro duction 1 1. Preliminarie s 4 1.1. Spectra l theory 4 1.2. Spaces of pro jectors 6 1.3. F redholm op erato rs and rela tive dimension 7 2. Essentially hyperb olic op erators 8 3. The s pectra l flow 12 3.1. Spectra l sections 14 4. Spec tr al flow as group homomo rphism 15 4.1. The Douady space 17 5. The F redholm index of F A and the sp ectral flow 19 5.1. The sp ectral flow and the F redho lm index of F A 21 References 22 ON THE SPECTRAL FLO W F O R P A T HS OF ESSENTIALL Y HYPERBOLIC BOUNDED OPERA TORS ON BANA CH SP ACES GARRISI D ANIELE T o my p ar e nts and my siste r Abstract. W e giv e a definition of the spectral flow for paths of bounded es- sen tially hyperb olic operators on a Banach space. The sp ectral flo w induces a group homomorphism on the fundament al group of every connected com- ponent of the space of essen tially hy perb olic operators. W e prov e that this homomorphism completes the exact homotop y sequence of a Serre fibration. This allows us to ch aracterise its kernel and image and to pr o duce examples of spaces where it is not injective or not surjective, unlike what happens f or Hilb ert spaces. F or a large class of paths, namely the essentially spli tting, the sp ectral flow of A coincides wi th − ind( F A ), the F redholm index of the differen tial op erator F A ( u ) = u ′ − Au . Introduction The sp ectral flo w first app eared in [ 7 ] for a family o f elliptic and self-adjoint op erators A t , ascrib ed to the joint work of M. Atiy ah and G. Lusztig. W e outline their effective description as “ ne t num b er o f eigenv alues that change sig n (from − to +) while the parameter family is completing a p erio d” in the definition given by J. Robbin and D. Salamon in [ 24 , Theore m 4.21 ]: In a ne ig h bo ur hoo d [ t − , t + ] of the real line of a p oint t ∈ R (called a crossing ) s uc h that 0 ∈ σ ( A t ) (the sp ectrum of 1991 Mathematics Subje ct Classific ation. 46B20,58B05,58E05,58 J30. Key wor ds and phr ases. Spectral flow, pro jectors, h yperplanes. This work was supp orted by the Pr iorit y Research Cen ters Program through the National Researc h F oundation of Korea (NRF) funded b y the Ministry of Education, Science and T ec hnology (Grant 2009-009406 8) and the Scuola Norm ale Sup eriore of Pisa. 1 2 GARRISI, D. A t ), σ ( A s ) can b e descr ibed as a finite family of c o n tin uously differentiable curves λ i : [ t − , t + ] → R such that λ ′ i ( t ) 6 = 0 . The co ntribution to the spectr al flow of a crossing is given by X λ i sign ( λ i ( t + )) − sig n ( λ i ( t − )) and the sp ectral flow is the s um of these contributions ov er all the crossing s. The sp ectral flo w was used to define a Mor se index for 1- perio dic hamiltonian orbits, as in [ 27 , pp. 2 3 ], and then to define the Flo er ho mology . The definition by J. Ro bbin and D. Salamon in [ 24 ] r e q uires s o me differentiabilit y h yp otheses and transversality conditions. P . Rabier extended their work to unbounded families of op erato rs in Banach spac e s in [ 23 ]. In [ 2 1 ], J. P hillips simplified their definition as follows: A t ∈ F sa ( H ) is a ssumed to b e a co n tin uous path of F re dholm, b ounded and self- adjoint op erator s, on [0 , 1 ]. If U is a neighbourho o d of the or igin and J = [ t − , t + ] is a closed interv a l such that 1) σ ( A s ) ∩ ∂ U = ∅ for every s ∈ J ; 2) σ ( A s ) ∩ U is a finite set of eigenv alues, then the contribution to the s pectra l flow fro m the interv al J is defined as dim( P ( A ( t + ); U )) − dim ( P ( A ( t − ); U )) where P ( A ; U ) is the s pectra l pro jector o f A relative to U . The s pectra l flow is the sum of all the contributions obtained over a par titio n of the unit interv al, J i , such that a neighbourho o d U i as in 1 ,2) corre s ponds to each J i . It is inv a riant for fixed-endp oin t homotopies a nd defines a groups homo morphism on the fundamen tal group of each connected comp onent of F sa ( H ). If H is infinite-dimensional and separable, then there ar e exac tly three co nnected comp onents, corr espo nding to I and − I (these are co n tractible to a p oin t) and to 2 P − I , wher e P is a pro jector with infinite-dimensional k ernel and ima ge. On the fundament al group of the third one, denoted by F sa ∗ , the sp ectral flow sf : π 1 ( F sa ∗ ) → Z is a g roup isomor phism. In [ 8 ], J. Phillips later extended this definition to co ntin- uous families of unbounded, F redholm and self-adjo int o pera tors. In contrast with [ 24 ], no ass umptions of differentiabilit y o r transversality are made. C. Zhu and Y. Long in [ 29 ] extended the definition in [ 2 1 ] to b ounded, admissible op erator s (whic h a re compact p erturbations of hyperb olic o pera tors) on B anach spaces : On every interv al J i = [ t i − , t i + ] of a s uitable partition of [0 , 1], they provide a path of pro jectors Q i such that Q i : J i → P ( L ( E )) is contin uous , Q i ( t ) − P + ( A ( t )) is compact for t ∈ J i , Q i ( t ) is a sp ectral pro jector o f A ( t ) . The c on tribuition of J i to the sp ectral flow is [ Q i ( t i − ) − P + ( A ( t i − ))] − [ Q i ( t i + ) − P + ( A ( t i + ))] , where [ Q − P ] denotes the F redholm index o f P : Range( Q ) → Range( P ). The sp ectral flow is defined as the sum o f the quantit y ab ov e as J i v aries ov er the partition o f [0 , 1]. W e denote the sp ectral pr o jector re la tiv e to the p ositive complex SPECTRAL F LOW IN BANACH SP ACES 3 half-plane by P + ( A ), and P ( L ( E )) is the set of pro jectors of E . This work has three essential purp oses: I) F ur ther simplifying the definition of sp ectral flow. Given a pa th A t ∈ e H ( E ) on [0 , 1 ] of essentially hyperb olic o pera tors (which a re compact per turbations o f hyperb olic op erators and co rresp ond to the admissible o nes used in [ 29 ]), there exists a contin uo us path P of pro jectors on [0 , 1] such that P ( t ) − P + ( A ( t )) is compa ct for every t ∈ [0 , 1]. Thus, we define sf ( A ) = [ P (0) − P + ( A (0))] − [ P (1 ) − P + ( A (1))] . Therefore, we do no t need to partition the unit interv al (as in the definition in [ 29 ]), as long as we do not req uir e P ( t ) to b e a sp ectral pro jector of A ( t ). The ex istence of such a path P follows from the ho motop y lifting prop erty o f the Serre fibration p : P ( L ( E )) → P ( C ( E )) . W e use C ( E ) to denote the quotient o f the oper ator alg ebra L ( E ) b y the ideal of the compact ope r ators. P ( C ( E )) is the s pace of pro jectors o f the Calkin alg ebra and p is the quotient pro jection. The definition we give of sp ectral flow coincides with the one in [ 29 ]. In § 3 , we show that the sp ectral flow is inv ar ian t for fixed-endp oin t homotopies and that, given t wo con tin uous paths A and B s uc h that A (1) = B (0), there holds sf ( A ∗ B ) = sf ( A ) + s f ( B ). These and other bas ic facts are o utlined in Prop osition 3.3 . II) Studying the homo morphism prop erties of the sp ectral flow. W e prove that e H ( E ) is homotopically equiv alent to P ( C ( E )) and that the spectr al flow completes the exact homo top y sequence of the fibratio n above. This allows us to characterise the kernel and the imag e of the sp ectral flow s f P defined o n the fundamental g roup of the co nnected co mponent o f 2 P − I ∈ e H ( E ). Precisely: an integer m b elongs to the image of sf P if and only if h1) ther e exists a pro jector Q connected to P b y a path in the space of pro- jectors P ( L ( E )), such that Q − P is compact and [ P − Q ] = m ; ker( sf P ) ∼ = im( p ∗ ). Hence, sf P is injectiv e if and o nly if h2) im ( p ∗ ) = { 0 } . The ex istence of a pro jector sa tisfying h1,2) depends heavily on the str ucture of the Bana c h space E . When E is Hilb ert, J. Phillips proved in [ 21 ] that for e v ery pro jector with infinite-dimens io nal r ange and kernel, h1 ), with m = 1 , and h2) hold. W e show that ℓ p and ℓ ∞ hav e this prop erty as well. In genera l, at least one pro jector (and then infinitely many) exists in the s paces s atisfying the hypotheses of P rop osition 4.6 and Pro p osition 4.7 . The question whether prop erty h1) holds for some pro jector is strongly related to the e xistence of co mplemen ted s ubspaces is omorphic to clo sed subspa ces of co- dimension m . This r elation is highlighted in P rop o sition 4.6 . In fa c t, given a space E , isomorphic to its hyperplanes, the pro jector ov e r each o f the summands of E ⊕ E fulfills pro per ty h1). Thu s, sf P is surjective. If E is iso morphic to its subspaces of co - dimension tw o, but not to its hyperpla nes, the image of the sp ectral flow is 2 Z , and so o n. E xamples of such spa ces have b een constructed by W. T. Gow e r s and B. Maurey in [ 15 ]. If E is not iso mo rphic to a n y of its prop er subspaces, then 4 GARRISI, D. sf P is zero . Such a space was constructed by W. T. Gow ers and B. Maurey in the celebrated pap e r [ 14 ]. W e prov e that in a Douady space (cf. [ 12 ]), there are pro jectors P such that sf P is not injective and pro jectors with infinite-dimensional ra ng e and kernel such that the sp ectral flow is zero. II I) Co mpa ring the sp ectral flow with the F redholm index of the differe ntial op erator F A : W 1 ,p ( R ) → L p ( R ) , u 7→  d dt − A ( t )  u. In § 5 , we extend the definition of sp ectral flow for a path A t ∈ e H ( E ) on R with hyperb olic limits at ±∞ . W e prov e in Theorem 5.6 that for a lar ge class of paths, which are essentially hype r bolic , with hyperb olic limits, and essentially splitting (cf. [ 2 ]), the equality ind( F A ) = − sf ( A ) holds. The e q ualit y ab ov e applies, for instance, in the s pecial ca se wher e A is a contin uous, compact p erturbation of a pa th of hyper bolic op erators with some bo undary conditions (ch eck [ 2 , T heo rem E]). Our theor em confirms the guess of A. Abbondando lo and P . Ma jer in § 7 of [ 2 ] that for these paths, the equality ab o ve holds. W e remar k that o ur work dea ls with b ounded op erators . The differential op er- ator F A arises naturally fr om the linear isation of a vector field ξ ∈ C 1 ( E , E ) o n a solution v ′ ( t ) = ξ ( v ( t )), such tha t the endp oints are zero es of ξ and A ( t ) = D ξ ( v ( t )). If F A is F redholm a nd surjective (a nd the zero es are hyperb olic), then the set W ξ ( p, q ) = { v : R → E : v ′ ( t ) = ξ ( v ( t )) , v ( −∞ ) = p, v (+ ∞ ) = q } is a sub-manifold of dimension ind( F A ). This constitutes a landmark fo r the study of the Mor se theor y on Banach manifolds , as in [ 3 ]. A pro of of this can b e found in [ 2 , § 8]. If A fulfils the hypo theses of Theorem 5 .6 , then sf ( A ) determines the dimension of the manifold. The sp ectral flow also provides an index for 1-p erio dic solutions o f u ′ ( t ) = X ( u ( t )). If D X ( u t ) is essentially h yper b olic, then index X ( u ) = sf ( DX ( u t )). Thus, in or der to hav e a go o d Mor se theory fo r p erio dic solutions, the question whether there are lo ops with non-triv ial sp ectral flow b ecomes relev a nt. Ackno wledgements . I would like to thank Pro f. Alber to Abb ondandolo and Prof. Pietro Ma jer for their aid a nd sug gestions, the r eferees for their ca reful reading and for po in ting out to me the pap ers of Yiming Lo ng and Z hu Chao feng . I also thank m y parents and my s is ter for a lw ays suppo rting me. 1. Preliminaries Here we review some basic definitions and results on sp ectral theory and F red- holm op erators. O ur main r eferences are [ 25 , Ch. X] and [ 16 , IV.4,5]. 1.1. Sp ectral theory. A Bana c h a lgebra is a n alg ebra A with unit, 1, over the real o r c o mplex field and a norm | | · | | such tha t  A , | | · | |  is a Bana ch space | | xy | | ≤ | | x | | · | | y | | for every x, y ∈ A . SPECTRAL F LOW IN BANACH SP ACES 5 we denote b y G ( A ) the set of inv er tible elements of the algebra; it is a n op en subset of A . Given x ∈ A , the subset of the field σ ( x ) = { λ ∈ F : x − λ · 1 6∈ G ( A ) } is called the sp ectrum of x . Let us re view so me pro p erties of the sp ectrum. Prop osition 1 .1. F or every x ∈ A , t ∈ F and Ω ⊂ F an op en subset, (i) if σ ( x ) is non-empty, then it is close d and b ounde d; (ii) σ ( x + t ) = σ ( x ) + t , σ ( tx ) = tσ ( x ) ; (iii) then ther e exists δ > 0 such t hat σ ( y ) ⊂ Ω for every y ∈ B ( x, δ ) ; (iv) if A is c omplex, then σ ( x ) is non-empty; (v) if f : A → B is an algebr as homomorphism su ch that f (1) = 1 , then σ ( f ( x )) ⊆ σ ( x ) . Given a real algebra, we c a n conside r the complex algebr a asso ciated with it, A C = A ⊗ R C . W e hav e an inclusio n of a lgebras A ֒ → A C , x 7→ x ⊗ 1 and σ ( x ) ⊆ σ ( x ⊗ 1). Hereafter , we will take σ ( x ⊗ 1) as the sp ectrum o f x . Definition 1.2. A finite family o f clos ed cur ves in the complex plane, Γ = { c i : 1 ≤ i ≤ n } , is sa id to b e simple if, fo r every z 6∈ S n i =1 im( c i ), ind Γ ( z ) := 1 2 π i Z Γ dζ z − ζ = 1 2 π i n X i =1 Z c i dζ z − ζ ∈ { 0 , 1 } . we denote by Ω 0 (Γ) a nd Ω 1 (Γ) the subsets of the complex space such that ind Γ ( z ) is 0 and 1, resp ectively . Definition 1.3 . An element p ∈ A is said to b e a pro jector if p 2 = p . W e can asso ciate with it the sub-a lg ebra A ( p ) = { p xp : x ∈ A} , with the unit p . we denote by σ p ( y ) the sp ectrum of an element y ∈ A ( p ). If xp = px , we hav e the e q ualit y (1) σ ( x ) = σ p ( xp ) ∪ σ 1 − p ( x (1 − p )) . Theorem 1.4. L et x ∈ A , Σ ⊂ σ ( x ) op en and close d in σ ( x ) . Then, ther e ex ists a pr oje ctor, c al le d a s p ectral pro jector r elative t o Σ , which we denote by P ( x ; Σ) , such that (i) σ P ( P xP ) = Σ ; (ii) P x = xP ; (iii) I − P = P ( x ; Σ c ) ; (iv) given an algebr a homomorphism, f : A → B , f ( P ( x ; Σ)) = P ( f ( x ); Σ ) . Given Γ simple su ch that Σ = Ω 1 (Γ) ∩ σ ( x ) , P ( x ; Σ) = P Γ ( x ) := 1 2 π i Z Γ ( x − ζ ) − 1 dζ . On the op en subset A (Γ) = { x ∈ A : σ ( x ) ∩ Γ = ∅} , P Γ ( x ) is a c ontinu ous map. F or the pro of and more details check [ 25 , Theore m 1 0.27] o r [ 16 , Theorem 6.1 7]. 6 GARRISI, D. 1.2. Spaces of pro jectors. Given B anach s paces E and F , we denote b y L ( E , F ) the space o f b ounded op erators from E to F . When E = F we use the notation L ( E ). W e deno te the subs e ts of compact op erators by L c ( E , F ) and L c ( E ). The comp osition of op erators endows the space L ( E ) with the str ucture o f a B anach a l- gebra (the identit y op erator b eing the unit), and the s ubspace o f compact op erators is a closed ideal. W e denote the quotient algebra by C ( E ). It inherits the structure of a Banach alge br a and is called the Calkin algebra . The quotient pro jection (2) p : L ( E ) → C ( E ) , A 7→ A + L c ( E ) is an algebra homomor phism. Definition 1. 5. Given a Banach algebra A , we define the following subsets (i) P ( A ) = { p ∈ A : p 2 = p } , pro jectors ; (ii) Q ( A ) = { q ∈ A : q 2 = 1 } , squar e ro ots of the unit ; (iii) H ( A ) = { x ∈ A : σ ( x ) ∩ i R = ∅} , hyperb olic elemen ts. Pr op erties and r emarks. P and Q ar e clo sed subsets, lo ca lly pa th connected and analytic sub- manifolds. A pro of of this c an b e found in [ 4 , Le mma 1.5 ]. P and Q are diffeomorphic to each other through the diffeomor phism p 7→ 2 p − 1. By (iii) of Prop osition 1.1 , applied with Ω = { z : Re( z ) 6 = 0 } , the subset H ( A ) ⊂ A is op en. W e denote by G 1 ( A ) the connected co mponent of G ( A ) of the unit. Theorem 1. 6. Given two pr oje ctors p, q such t hat either | | p − q | | < 1 or b oth ar e in the same c onne cte d c omp onent of P ( A ) , ther e exists u ∈ G 1 ( A ) such that up = q u . F or the pro of and details, we refer to [ 22 , Pro positio n 4.2] and [ 13 , Pr o pos i- tion 2.2]. The theorem ab ov e has tw o consequences: (c1) P ( A ) is lo cally path-connected. So is Q ( A ); (c2) when A = L ( E ), tw o pro jectors in the same connec ted comp onen t hav e isomorphic ra nges and kernels. The quo tient pro jection p in ( 2 ) r e stricts to the subset o f pro jectors and ro ots o f the unit P ( p ) : P ( L ( E )) → P ( C ( E )) , P 7→ P + L c ( E ) Q ( p ) : Q ( L ( E )) → Q ( C ( E )) , Q 7→ Q + L c ( E ) . Definition 1.7. A contin uous map p : E → B has the ho motop y lifting pr oper t y w.r.t. a top ological space X if, given contin uo us maps h : X × [0 , 1] → B , f : X × { 0 } → E , there exists H : X × [0 , 1] → E s uc h that H ( x, 0) = f ( x, 0) a nd p ◦ H = h . If the homotopy lifting pro perty holds w.r.t. [0 , 1] n for every n ≥ 0, then p is called a Serre fibration . Prop osition 1 .8. The maps P ( p ) and Q ( p ) ar e surje ctive Serr e fibr ations. In g e neral, e very s urjectiv e algebr a homomor phis m induces a Serre fibration. F or a pro of, see [ 10 , Theorem 2.4]. T he sur jectivity o f P ( p ) and Q ( p ) follows fro m [ 4 , Prop osition 4.1]. In fact, P ( p ) and Q ( p ) are lo cally trivia l fib er bundles, as follows from [ 4 , Prop osition 1.3] or [ 13 , Theor e m 4.2 ]. SPECTRAL F LOW IN BANACH SP ACES 7 1.3. F redhol m op erators and rel ativ e dim ension. Let T ∈ L ( E , F ) b e a bo unded o pera tor. If the imag e of T is a clo sed subspace, we have tw o B a nach spaces asso ciated with it, namely ker( T ) and E / Range( T ) = coker( T ). Definition 1 .9. An op erator as ab ov e is sa id to be semi-F redho lm if either ker( T ) or co k er( T ) is a finite-dimensional space. If bo th hav e finite dimensio n, T is called F redholm and the integer ind( T ) = dim ker ( T ) − dim co k er( T ) is the F redholm index . Otherwise, the index is defined to be + ∞ or −∞ as long as ker( T ) or coker( T ) has infinite dimension. we de no te by F red( E , F ) and F red( E ) the subse ts of F re dholm op erators in L ( E , F ) and L ( E ), resp ectively; F red k ( E , F ) is the set of F redholm o pera tors of index k . Prop osition 1.10. L et T ∈ F red( E , F ) , S ∈ F red( F , G ) and K ∈ L c ( E , F ) . We have (a) F red k ( E , F ) ⊆ L ( E , F ) is an op en subset; (b) T + K ∈ F red( E , F ) and ind( T + K ) = ind ( T ) ; (c) S ◦ T ∈ F r ed( E , G ) and ind( S ◦ T ) = ind ( S ) + ind( T ) ; (d) given B ∈ L ( E , F ) , ther e exists ε > 0 s u ch t hat the maps dim ker ( T + λB ) and dim coker( T + λB ) ar e c onstant on B (0 , ε ) \ { 0 } ; (e) T ∈ F red( E , F ) if and only if ther e exists U ∈ L ( F , E ) such that T and U ar e the essential in verse of e ach other, that is, T ◦ U − I ∈ L c ( F ) and U ◦ T − I ∈ L c ( E ) . Statement s (a,e) ar e eas y to c heck. Statements (a,b,d) are all sta ted a nd prov ed in [ 16 , Ch. I V.5 ] in the more gener al setting o f semi-F redho lm and unbounded op erators. Definition 1.11. A pair of closed subspac e s ( X , Y ) is semi-F redholm if and only if their sum is closed a nd either X ∩ Y o r E / ( X + Y ) has a finite dimension. If bo th hav e finite dimension, then the F redholm index of the pair ( X , Y ) is defined as ind( X, Y ) = dim X ∩ Y − co dim X + Y . Otherwise, the index is + ∞ or −∞ , when either X ∩ Y o r E / ( X + Y ) has infinite dimension. Two pro jectors P , Q are co mpa ct pe r turbations o f each o ther if P − Q ∈ L c ( E ). In this ca se, the restriction of Q to Range( P ) is in F r ed(Range( P ) , Rang e( Q )). T he relative dimension b etw een P a nd Q is defined as [ P − Q ] := ind( Q : Range( P ) → Range( Q )) . This definition is meant to generalise the dimension g ap b etw een tw o finite-dimensional spaces to Banach spaces. The notation ab ov e is used b y C. Zhu and Y. Long in [ 29 ]. Corresp onding definitions are k nown in Hilb ert spaces, considered by A. Abbo n- dandolo and P . Ma jer in [ 1 , Definition 1.1 ] (see als o [ 9 , Remark 4.9]). A definition of r elative dimens io n fo r pair s of closed subspaces ( X , Y ), not necessa rily comple- men ted, can b e found in [ 13 , Definition 5.8]. Theorem 1.12. Given p airs of pr oje ctors ( P, Q ) and ( Q, R ) with c omp act differ- enc e, we have 8 GARRISI, D. (i) if Range( P ) and Range( Q ) have finite dimension, t hen [ P − Q ] = dim Rang e( P ) − dim Range( Q ) ; (ii) [ P − R ] = [ P − Q ] + [ Q − R ] ; (iii) on the subset { ( P, Q ) ∈ P ( L ( E )) × P ( L ( E )) : P − Q ∈ L c ( E ) } , the map [ P − Q ] is c ontinuous; (iv) [ P − Q ] = [( I − Q ) − ( I − P )] ; (v) (Range( P ) , ker( Q )) is a F r e dholm p air and ind(Rang e( P ) , ker( Q )) = [ Q − P ] . Prop erty (iii) follows from stability res ults for the index of semi-F r edholm pa ir s; see [ 16 , Rema rk IV.4.31 ] and [ 13 , Theor em 3 .3]. F or a pro of o f (iv) and (v), see [ 2 9 , Lemma 2.3] and [ 13 , P rop osition 5.1 3], resp ectiv ely; (i) follows from the rema rks after Definition 5.8 in [ 1 3 ]. 2. Essentiall y hyperbolic opera tors W e recall that a bo unded op erator A ∈ L ( E ) - or more g enerally an ele ment o f a Banach algebr a A - is ca lled h yper bolic if its sp ectrum do es not meet the ima g inary axis. W e denote by GL ( E ) the group of in vertible op erators on E and by GL I ( E ) the connected comp onent of the identit y op erator. Given A ∈ L ( E ), the sp ectrum of A + L c ( E ) is called the essential sp ectrum. It is usually denoted by σ e ( A ). By (e) of Pr opo sition 1.10 , (3) σ e ( A ) = { λ : A − λ 6∈ F red( E ) } . Definition 2. 1. An op erator A is called e ssen tially hyperb olic if A + L c ( E ) is a hyperb olic element in C ( E ). By the equa lit y ab ove, an op era to r A ∈ L ( E ) is essentially hyperb olic if and only if its essential sp ectrum do es not meet the imaginar y axis. A co nsequence o f ( 3 ) is: Lemma 2.2 . L et D ( A ) b e the set of al l isolate d p oints of σ ( A ) , and let ∂ σ ( A ) b e the set of the b oundary p oints of σ ( A ) . Then ∂ σ ( A ) \ D ( A ) is a subset of σ e ( A ) . Pr o of. Let λ ∈ ∂ σ ( A ) \ D ( A ), a nd suppo se that λ 6∈ σ e ( A ), thus A − λ ∈ F r ed( E ). Let ε > 0 a s in (d) of Pr o pos ition 1.10 , with B = − I . Therefor e, for so me c, k ∈ Z dim ker ( A − z ) = c, dim coker( A − z ) = k , for every z ∈ B ( λ, ε ) \ { λ } . Because λ ∈ ∂ σ ( A ), there exists w ∈ B ( λ, ε ) \ { λ } such that A − w is inv e r tible. Thu s, c = k = 0 and A − z is invertible fo r every z ∈ B ( λ, ε ) \ { λ } . Hence λ is isolated in σ ( A ).  W e need a well-known fact ab out the top ology of the rea l line: Prop osition 2. 3 . A close d pr op er subset of the r e al line with an empty b oundary is discr ete. Corollary 2.4. If A is an essential ly hyp erb olic op er ator, the set σ ( A ) ∩ i R is finite. Pr o of. W e s ho w that the b oundary of σ ( A ) ∩ i R is empty . Supp ose it is not and let λ ∈ ∂ ( σ ( A ) ∩ i R ) b e an arbitrar y p oin t. Hence, λ ∈ ∂ σ ( A ). Because σ ( A ) ∩ i R is closed, λ ∈ i R . Hence λ 6∈ σ e ( A ), b e cause A is ess en tially hyper bolic. Thus, λ ∈ ∂ σ ( A ) \ σ e ( A ), whence, by Lemma 2.2 , λ ∈ D ( A ). Hence, λ is isola ted in σ ( A ) ∩ i R in contradiction with the hypothesis that λ is a b oundar y p oint. By the SPECTRAL F LOW IN BANACH SP ACES 9 prop osition ab ov e, σ ( A ) ∩ i R is dis crete. Because it is also compact, it is a finite set.  Prop osition 2. 5 . If A is an essential ly hyp erb olic op er ator, e ach of the p oints of σ ( A ) ∩ i R is an eigenvalue of fin it e algebr aic multiplicity. Pr o of. Let λ ∈ σ ( A ) ∩ i R . W e infer that A − λ ∈ F red 0 ( E ). By ( 3 ), A − λ ∈ F red k ( E ) for some k ∈ Z . Now, by (a ) of Prop osition 1.10 , ther e exis ts a neighbourho o d V of λ such that A − z ∈ F red k ( E ) , z ∈ V . Because λ is isolated, there ex ists z ′ ∈ V \ { λ } suc h that A − z ′ is inv ertible, hence k = 0. Thus, b ecause A − λ is not inv ertible, ker( A − λ ) 6 = { 0 } . Hence λ is an eigenv alue and, by hypothes is , isolated. These t w o conditions, by Theorem 5.10 and Theorem 5.28 o f [ 16 ], imply that the sp ectral pro jector P ( A ; { λ } ) has range of finite dimension, which is the a lgebraic multiplicit y .  Theorem 1.4 provides us with pro jectors P i = P ( A ; { λ i } ) for ev ery λ i ∈ σ ( A ) ∩ i R . Let P = P ( A ; σ ( A ) ∩ { Re( z ) 6 = 0 } ). W e can write (4) A = AP + n X i =1 P i ! + ( A − I ) n X i =1 P i . According to (i,ii) of Theorem 1.4 , the term in the brack ets is hyp e rbo lic . The last term has finite ra nk. Th us , we hav e proved that an essentially hyperb olic op erator is a compact p erturba tion of a hyperb olic one. Con v ersely , a compa c t per turbation of a hyper bolic op erator is essentially hyperb olic. In fact, let H , K be a hyperb olic and a compac t op erator , resp ectively: By (b) of Pro positio n 1.10 and ( 3 ), σ e ( H + K ) = σ e ( H ) ⊆ σ ( H ). Because H is h yp erbo lic, σ ( H ) do es not meet the imaginary axis , so neither do es σ e ( H ). Therefore, H + K is essentially hyperb olic. Thus, by ( 4 ) and the rema rks after it, we have prov ed the following Theorem 2.6. An op er ator is essent ial ly hyp erb olic if and only if it is a c omp act p erturb ation of a hyp erb olic op er ator. we denote by e H ( E ) the set of es s en tially hype r bolic op erator s endow ed with the top ology induced by the op erator norm. Prop osition 2.7. e H ( E ) is an op en subset of L ( E ) , and home omorph ic to the pr o duct H ( C ( E )) × L c ( E ) . Pr o of. By Definition 2.1 , (5) e H ( E ) = p − 1 ( H ( C )) , where p is the q uotien t pr o jection defined in ( 2 ). Because the rig ht term is an op en s ubset of L ( E ), s o is e H ( E ). Because p is linear, contin uous and surjective, there exists s : C ( E ) → L ( E ) contin uous such that p ◦ s = id . This follows from [ 4 , Prop osition A.1]. W e define the contin uous maps f : e H ( E ) → H ( C ( E )) × L c ( E ) , A 7→ ( p ( A ) , A − s ( p ( A ))) , g : H ( C ( E )) × L c ( E ) → e H ( E ) , ( x, K ) 7→ s ( x ) + K. Both are well defined. In fact, by ( 5 ), p ( A ) ∈ H ( C ( E )). By the pr oper t y of s , p ( A − s ( p ( A ))) = 0 , 10 GARRISI, D. th us the L ( E ) comp onent of f is compact, b ecause L c ( E ) = ker( p ). As for g , bec ause p ( s ( x )) = x ∈ H ( C ( E )), by ( 5 ), s ( x ) ∈ e H ( E ), hence σ e ( s ( x )) ∩ i R = ∅ . Because σ e ( s ( x ) + K ) = σ e ( s ( x )), s ( x ) + K ∈ e H ( E ). W e conclude the pro of by chec king that f and g are the inv ers es of each o ther: f ◦ g ( x, K ) = f ( s ( x ) + K ) = ( p ( s ( x ) + K ) , s ( x ) + K − s ( p ( s ( x ) + K ))) = ( x, K ) g ◦ f ( A ) = s ( p ( A )) + A − s ( p ( A )) = A.  Definition 2.8. Let x ∈ A b e such that σ + ( x ) = σ ( x ) ∩ { Re( z ) > 0 } is op en and closed in σ ( x ). W e deno te b y p + ( x ) the pro jector P ( x ; { Re( z ) > 0 } ). Simila rly , we define p − ( x ). Prop osition 2.9. The map p + : H ( A ) → P ( A ) defines a homotopy e quivalenc e, a homotopy inverse b eing the map j : P ( A ) → H ( A ) , j ( p ) = 2 p − 1 . Pr o of. F or x ∈ H ( A ), σ + ( x ) is o pen and closed in σ ( x ). Thus, there exists a rectangle Q := ( − a, a ) × ( − b , b ) s uc h that σ ( x ) ⊂ Q . There is a contin uous, closed and simple (in the sense of Definiton 1.2 ) curve c , suc h that im( c ) = ∂ Q + . Thu s, p + ( x ) = P c ( x ). By (iii) of P rop osition 1.1 , there exists δ > 0 such that, if d ( y , x ) < δ , then y ∈ H ( A ) and σ ( y ) ⊂ Q . Thu s Q + ∩ σ ( y ) = σ + ( y ) and p + ( y ) = P c ( y ). By Theor e m 1.4 , P c is co n tin uous o n B ( x, δ ). Thus, p + is co n tin uous o n a neighbourho o d of x , namely B ( x , δ ). Repea ting the sa me arg umen t for every x , we obtain that p + is contin uous on H ( A ). Given a pro jector p , j ( p ) is a sq ua re ro ot of the unit. In fact, j ( p ) 2 = (2 p − 1 ) 2 = 4 p 2 − 4 p + 1 = 1 . Thu s σ ( j ( p )) ⊆ {− 1 , 1 } , hence j ( p ) is hyp e rbo lic . Given ζ ∈ C \ {− 1 , 1 } , we hav e ( j ( p ) − ζ ) − 1 = ζ 1 − ζ 2 + j ( p ) 1 − ζ 2 = 1 2  − 1 ζ + 1 + 1 1 − ζ  − 1 2  − 1 ζ + 1 − 1 1 − ζ  j ( p ) . Let c b e a simple curve a s c ( t ) = 1 + e − 2 π it / 2. Thus, following the notations o f Definition 1.2 , we hav e σ + ( j ( p )) = σ ( j ( p )) ∩ Ω 1 ( c ) . Therefore, we can compute the sp e ctral pro jector r elative to 1 ∈ σ ( j ( p )) as in Theorem 1 .4 . By in tegrating b oth sides of the ab ov e equality , p + ( j ( p )) = 1 2 π i Z c ( j ( p ) − ζ ) − 1 dζ = 1 2  − ind c ( − 1) + ind c (1)  − 1 2  − ind c ( − 1) − ind c (1)  j ( p ) = 1 2  0 + 1 + j ( p )  = 1 2 (1 + 2 p − 1) = p. The co mputation ab ov e shows that p + ◦ j is the ident it y map on P ( A ). T o prov e that j ◦ p + is homotopica lly equiv alent to the identit y o n H ( A ), we define (6) H ( t, x ) =  (1 − t ) x + t  p + ( x ) +  (1 − t ) x − t  p − ( x ) . SPECTRAL F LOW IN BANACH SP ACES 11 Because x is hyperb olic, σ + ( x ) ∪ σ − ( x ) = σ ( x ), thus, by (iii) of Theorem 1.4 , (7) p + ( x ) + p − ( x ) = 1 . By ( 1 ) and by (ii) of Theo rem 1 .4 , we have σ ( H ( t, x )) = σ p + ( x )  (1 − t ) xp + ( x ) + tp + ( x )  ∪ σ p − ( x )  (1 − t ) xp − ( x ) − tp − ( x )  = { (1 − t ) σ + ( x ) + t } ∪ { (1 − t ) σ − ( x ) − t } . (8) The s econd eq ualit y follows fro m (i) of Theorem 1.4 a pplied to p + ( x ) (resp. p − ( x )) and σ + ( x ) (res p. σ − ( x )). B e cause the subsets of the complex plane { Re( z ) > 0 } and { Re( z ) < 0 } are conv ex, the sets in the se cond line of ( 8 ) do not meet the imaginary axis, and thus H ( t, x ) is hype r bolic . Mo reov er , H (0 , x ) = p + ( x ) + p − ( x ) = 1 H (1 , x ) = p + ( x ) − p − ( x ) = 2 p + ( x ) − 1 = j ( p + ( x )) by ( 7 ). Hence H is a homotopy of j ◦ p + with the identit y map.  Because L c ( E ) is a vector s pa ce, thus it is contractible to a p oint, the pro jection onto the firs t facto r in H ( C ( E )) × L c ( E ) is a homotopy equiv alence. T o gether with the last tw o pr o pos itions, we hav e pr oved the following Corollary 2.1 0. The map Ψ : e H ( E ) → P ( C ( E )) , A 7→ p + ( A + L c ( E )) is a ho- motopy e quivalenc e. Given a n essentially hyperb olic o p erato r A , we denote by P + ( A ) and P − ( A ) the sp ectral pro jectors relative to { Re( z ) > 0 } and { Re( z ) < 0 } , resp ectively . Prop osition 2.11. Given a c onne cte d c omp onent X ⊂ e H ( E ) , ther e exists P ∈ P ( L ( E )) such that 2 P − I ∈ X . Mor e over, t wo essent ial ly hyp erb olic op er ators A, B b elong t o t he same c onne cte d c omp onent X , if and only if t her e exists T ∈ GL I ( E ) such that T P + ( A ) T − 1 − P + ( B ) ∈ L c ( E ) . Pr o of. By Prop osition 2.7 , e H ( E ) is an op en subset o f L ( E ). Thus, X is path- connected. Let A ∈ X ⊂ e H ( E ) b e a n es sen tially hyperb olic ope rator. B y T heo - rem 2.6 , there exists a h yp erbo lic op erato r H such that A − H ∈ L c ( E ) . By the conv erse of the s a me theor em, the co n tin uous co n v ex co mbination γ : t 7→ A + t ( H − A ) lies in e H ( E ). Because γ (0) = A ∈ X , γ (1 ) = H ∈ X . By P rop o sition 2.9 , j ◦ p + ( H ) = 2 P + ( H ) − I is path-co nnec ted to H , a path b eing defined as in ( 6 ). Thu s 2 P + ( H ) − I ∈ X , and this concludes the first part of the prop osition. Given A, B ∈ X , there exists a path A t such that A (0) = A and A (1) = B . Thu s, the path α := Ψ ◦ A ∈ P ( C ( E )) connects α (0) = Ψ ( A ) to α (1) = Ψ( B ). By Prop osition 1.8 , the fibration ( P ( L ( E ) , P ( C ( E ) , p ) satisfies the homotopy lifting prop erty w.r.t. to the unit interv al [0 , 1]. Thus, there exists a path of pro jectors P such that (9) P (0) = P + ( A ) , p ( P ( t )) = α ( t ) . 12 GARRISI, D. By Theorem 1.6 , ther e exists T ∈ GL I ( E ) s uc h that (10) T P + ( A ) T − 1 = P (1) . F rom ( 9 ) with t = 1, we obta in p ( P (1)) = Ψ( B ) = p + ( B + L c ( E )) = P + ( B ) + L c ( E ) = p ( P + ( B )) . The second equality follows fro m the definition Ψ a nd the third one from (iv) of P rop osition 1.4 . Thus, co mpa ring the first and the last terms in the chain of equalities ab ov e, we obtain P (1) − P + ( B ) ∈ L c ( E ) . Hence, by ( 10 ), T P + ( A ) T − 1 − P + ( B ) ∈ L c ( E ) .  3. The spectral flow Let A : [0 , 1] → e H ( E ) b e a contin uous path. By Pro pos ition 2.10 , Ψ( A ( t )) is a contin uous path in P ( C ( E )). This path can b e lifted to a path o f pr o jector s P , such that p ( P ( t )) = Ψ( A ( t )) = p ( P + ( A ( t ))) by Prop osition 1.8 . W e define the in teger (11) sf ( A ; P ) := [ P (0) − P + ( A (0))] − [ P (1 ) − P + ( A (1))] . Prop osition 3.1. The inte ger sf ( A ; P ) do es n ot dep en d on the choic e of the p ath of pr oje ctors P . Pr o of. Let Q b e a path of pro jectors such that p ( Q ( t )) = p ( P ( t )). Thus, Q (0) − P (0) and Q (1) − P (1 ) a re compact op erators. By (ii) o f Theorem 1.12 , we hav e sf ( A ; Q ) = [ Q (0) − P + ( A (0))] − [ Q (1) − P + ( A (1))] = [ Q (0 ) − P (0)] + [ P (0) − P + ( A (0))] − [ Q (1) − P (1)] − [ P (1) − P + ( A (1))] = sf ( A ; Q ) By (iii) of Theo rem 1.12 , [ Q ( t ) − P ( t )] is constant. Thus, the third equalit y follows.  Definition 3.2. Given A : [0 , 1 ] → e H ( E ) contin uous, w e define the sp e c tral flow as the integer sf ( A ; P ) where P is any of the paths of pro jectors such that p ( P ( t )) = p ( P + ( A ( t ))). W e denote it by s f ( A ). Given T ∈ L ( E ) and S ∈ L ( F ), we refer to T ⊕ S as the linear op erator on E ⊕ F such that T ⊕ S ( x, y ) = ( T x, S y ). Given t w o paths A and B such that A (1) = B (0), we denote by A ∗ B the contin uous path A ∗ B ( t ) = ( A (2 t ) 0 ≤ t ≤ 1 / 2 B (2 t − 1) 1 / 2 ≤ t ≤ 1 Prop osition 3 .3. The sp e ctr al flow satisfies t he fol lowing pr op erties: (i) Given two p aths A and B su ch that A (1) = B (0) , sf ( A ∗ B ) = sf ( A ) + sf ( B ) ; (ii) the sp e ctra l flow of a c onstant p ath or a p ath in H ( L ( E )) is zer o; (iii) it is invariant for homotopies with endp oints in H ( L ( E )) and for fixe d- endp oint homotopies in e H ( E ) ; SPECTRAL F LOW IN BANACH SP ACES 13 (iv) if A i ∈ C ([0 , 1] , e H ( E i )) for 1 ≤ i ≤ n , then sf ( ⊕ n i =1 A i ) = P n i =1 sf ( A i ) ; (v) if E is an n -dimensional line ar sp ac e, then for every inte ger − n ≤ k ≤ n , ther e is a p ath such that s f ( A ) = k ; (vi) if E has infin ite dimension, then for every k ther e is A such that sf ( A ) = k . Pr o of. (i). Let A, B be t wo paths such that A (1) = B (0). W e ca n cho ose paths of pro jectors P and Q such that p ( P ( t )) = p ( P + ( A ( t )) and p ( Q ( t )) = p ( P + ( B ( t ))), with Q (0) = P (1). Denote by C and R the paths A ∗ B a nd P ∗ Q , r espectively . Then, sf ( A ∗ B ) = [ R (0) − P + ( C (0))] − [ R (1) − P + ( C (1))] = [ P (0 ) − P + ( A (0))] − [ Q (1) − P + ( B (1))] = [ P (0) − P + ( A (0))] − [ P (1) − P + ( A (1))] + [ Q (0 ) − P + ( B (0))] − [ Q (1) − P + ( B (1))] = sf ( A ) + sf ( B ) . (ii). If A is constant, P + ( A ( t )) is constant; if A is hyp erbo lic, P + ( A ( t )) is contin- uous. In b oth cas es, P + ( A ( t )) is a contin uous path and c an b e chosen as a lifting path o f p ( P + ( A ( t ))). Therefore , sf ( A ) = [ P + ( A (0)) − P + ( A (0))] − [ P + ( A (1)) − P + ( A (1))] = 0 . (iii). Let H : I × I → e H ( E ) be a contin uous map. By the homoto p y lifting prop erty of the fibre bundle p : P ( L ( E )) → P ( C ( E )) w.r .t. I 2 , ther e e xists P : I × I → P ( L ( E )) s uch that P ( t, s ) − P + ( H ( t, s )) ∈ L c ( E ) , for e very t, s. Let H ( · , 0) = A a nd H ( · , 1) = B . W e hav e sf ( A ) = [ P (0 , 0) − P + ( H (0 , 0))] − [ P (1 , 0 ) − P + ( H (1 , 0))] . F or i = 0 , 1 and every s , the o pera tor P ( i, s ) − P + ( H ( i, s )) is compact. F or a fixed i , the right s umma nd is co nstan t or co n tin uous, whether the ho motop y ha s fixed endpo in ts in e H ( E ) or lies in H ( L ( E )). In b oth ca ses, is co n tin uous. By (iii) o f Theorem 1.12 , there ar e integers k 1 , k 2 such that [ P ( i, s ) − P + ( H ( i, s ))] = k i for every 0 ≤ s ≤ 1 a nd i = 0 , 1. Thus, sf ( A ) = k 0 − k 1 = sf ( B ). (iv). Let P i be c o n tin uous paths of pro jectors such that P i ( t ) − P + ( A i ( t )) ∈ L c ( E i ). sf ( ⊕ n i =1 A i ) =  ⊕ n i =1 P i (0) − ⊕ n i =1 P + ( A i (0))  −  ⊕ n i =1 P i (1) − ⊕ n i =1 P + ( A i (1))  = n X i =1 [ P i (0) − P + ( A i (0))] − [ P i (1) − P + ( A i (1))] = n X i =1 sf ( A i ) . (v). W e denote the identit y on R k by I k . Given 0 ≤ k ≤ n , the s p ectral flow o f A ( t ) = (2 t − 1) I k ⊕ I n − k can b e computed using P ( t ) ≡ I n . Becaus e P + ( A (1)) = I n and P + ( A (0)) = 0 ⊕ I n − k , we hav e sf ( A ; I n ) = [ I n − 0 ⊕ I n − k ] − [ I n − I n ] = k by (i) of Theorem 1.12 . W e define A ( t ) := A (1 − t ). By prop erty (i), proved ab ov e , sf ( A ; I n ) = − k . 14 GARRISI, D. (vi). Given k ∈ Z , let E = X k ⊕ R k where X k is a clos e d s ubspace and dim( R k ) = k . Thu s, the sp ectral flow of A ( t ) = (2 t − 1) I R k ⊕ I X k ca n b e computed with P ( t ) ≡ I . W e obtain sf ( A ; I ) = k and sf ( A ; I ) = − k .  3.1. Sp ectral sections. The definition of sp ectral flow we used co rresp onds to the o ne given b y C. Zhu and Y. Lo ng in [ 29 ] for paths of admissible ope rators (see [ 29 , Definition 2 .3]), which are es sen tially hyperb olic. W e r e call the definition of s-section: Definition 3.4 . An s -section for a path of pro jectors Q on J ⊂ [0 , 1] is a contin uous path P such that P ( t ) − Q ( t ) ∈ L c ( E ). Given a con tin uous path A : [0 , 1 ] → e H ( E ), the authors s ho w in [ 29 , Lemma 2.5] and [ 29 , Corolla ry 2.1] that there e xists a partitio n o f the unit interv al ( J k ) n k =1 and P k : J k → P ( L ( E )) such that P k is an s-sec tio n for P + ( A ) on J k (12) P k ( t ) is a sp ectral pro jector o f A ( t ) . (13) Then, they define (14) sf ( A ) = n X k =1 sf ( A k ; P k ) where A k is the restr iction o f A to J k . In our definition we do not need to par- tition the unit in terv al b ecause w e dropp ed the r equirement ( 13 ). This a llows us to simplify the definition of sp ectral flow and to pr o vide simpler pr o ofs of well known prop erties - such a s the ho motopy inv aria nce - than the or iginal ones in [ 29 , Prop osition 2.2] o r in [ 21 ]. W e conclude by showing that there exists a path A such that P + ( A ( t )) do es not admit a n s-section fulfilling ( 13 ). Example 3.5. Cons ider the deco mp osition E = R 1 ⊕ X − ⊕ X + , wher e E is a Ba- nach space, X − and X + are closed, infinite-dimensio nal subspaces and dim( R 1 ) = 1 . Denote by P 1 , P − , P + the pr o jector s onto R 1 , X − and X + , resp ectively . Define A : [0 , 1] → e H ( E ) , A ( t ) = P + − P − + (2 t − 1) P 1 . Then A ( t ) ∈ e H ( E ), a nd no contin uous s-section satisfying ( 13 ) exists . Pr o of. F or every t ∈ [0 , 1] we can write A ( t ) = P + − ( P − + P 1 ) + 2 tP 1 ; beca use P + − P − − P 1 ∈ H ( L ( E )) (in fact, is a squar e ro ot of the iden tit y) and P 1 is compact, A ( t ) ∈ e H ( E ). By contradiction, supp ose that such P exists . W e have A (0) = P + − P − − P 1 , P + ( A (0)) = P + , σ ( A (0)) = { − 1 , 1 } . Because P (0) is sp ectral, there exists Σ 0 ⊂ σ ( A (0)) such that P (0) = P ( A (0); Σ 0 ). The o nly choice is Σ 0 = { 1 } , thus P (0) = P ( A (0); { 1 } ) = P + . On the other endpo in t, A (1) = P + − P − + P 1 , P + ( A (1)) = P + + P 1 , σ ( A (1)) = { − 1 , 1 } . As ab ov e, P (1) is sp ectral and P (1) = P ( A (1); { 1 } ) = P + + P 1 . Becaus e P is an s-section and P + ( A ( t )) − P + ( A ( s )) is compact for every 0 ≤ t, s ≤ 1, P ( t ) − P ( s ) is also compact. By (iii) o f Theo rem 1.12 , m ( t ) := [ P ( t ) − P (0)] is constant. Because m (0) = 0, m (1) = 0 . But, 0 = m (1 ) = [ P (1) − P (0)] = [( P + + P 1 ) − P + ] = [ P 1 ] = 1 SPECTRAL F LOW IN BANACH SP ACES 15 where the las t e qualit y follows from (i) of Theo rem 1.12 . Thu s, w e obtained a contradiction.  4. Spectral flow as group homomorphism By (iii) a nd (i) of P r opo sition 3.3 , the spec tr al flow determines a Z -v alued g roup homomorphism on the fundamental group of each co nnected comp onen t of e H ( E ). Given a pro jector P , we denote by sf P the sp ectral flow on the fundament al group of the connected comp onent of 2 P − I . The fib er of p : P ( L ( E )) → P ( C ( E )) ov e r a p oint o f the base space, P + L c ( E ) is the set P c ( E ; P ) = { Q ∈ P ( L ( E )) : Q − P ∈ L c ( E ) } i : P c ( E ; P ) ֒ → P ( L ( E )) . Prop osition 4. 1. F or every pr oje ct or P , the c onne cte d c omp onents of P c ( E ; P ) c orr esp ond to Z thr ough the bije ction Q 7→ [ P − Q ] . Mor e over, if the r ange and the kernel have infinite dimension, π 1 ( P c ( E ; P ) , P ) ∼ = Z 2 . The tw o facts follow from [ 13 , Theorem 6.3 ] and [ 13 , Theo rems 7.2,7 .3]. Because p induces a Ser re fibr ation, the sequence of homomorphis ms (15) π 1 ( P c ( E ; P ) , P ) i ∗ / / π 1 ( P ( L ( E )) , P ) p ∗ / / π 1 ( P ( C ( E )) , P + L c ( E )) is exact. The homoto py equiv alence Ψ defined in Coro llary 2.10 determines a group isomorphism Ψ ∗ : π 1 ( e H ( E ) , 2 P − I ) → π 1 ( P ( C ( E )) , P + L c ( E )) . Theorem 4 . 2. Ther e exists a homomorphism δ P such that t he se quenc e (16) π 1 ( P ( L ( E )) , P ) p ∗ / / π 1 ( P ( C ( E )) , P + L c ( E )) δ P / / Z , is exact and δ P ◦ Ψ ∗ = sf P . Pr o of. Let a b e a lo op at the base p oint P + L c ( E ), and let Q b e a path of pro jectors such tha t Q (0) = P and p ( Q ( t )) = a ( t ). Because a (0) = a (1), Q (1) − P is compact. W e define δ P ([ a ]) = [ Q (1) − P ] . Arguing as in Pr opo sition 3.1 , δ P is well defined. Let A b e a clo sed path in e H ( E ) and Q a path of pro jectors such that p ( Q ( t )) = Ψ( A ( t )) , Q (0) = P . Hence, Q ( t ) − P + ( A ( t )) is compact for every t ∈ [0 , 1]. By ( 11 ), sf ( A ) = [ Q (1 ) − P ]. By the definition ab ov e , δ P ([Ψ ◦ A ]) = [ Q (1 ) − P ], th us δ P ◦ Ψ ∗ = sf P . Because Ψ ∗ is in vertible, δ P is a homomor phism. W e prove tha t δ P is exact. ker( δ P ) ⊆ im( p ∗ ): Let a be a lo op at the base po in t P + L c ( E ) such that [ a ] ∈ ker( δ P ). Then, [ Q (1) − P ] = 0 . By Prop osition 4.1 , P a nd Q (1) ar e in the s ame co nnected c omponent in P c ( E ; P ). Thu s, there exists a co n tin uous path o f pro jectors R s uch tha t R (0) = Q (1) , R (1) = P, R ( t ) − P ∈ L c ( E ) for every t. 16 GARRISI, D. Set S := Q ∗ R . It is a closed path of pr o jecto r s, and p ◦ S = a ∗ c p ( P ) , wher e c p ( P ) is the constant path p ( P ). Thus, [ a ] ∈ im( p ∗ ). im( p ∗ ) ⊆ ker( δ P ): Given a lo op P t ∈ P ( L ( E )) at the base p oin t P , we hav e δ P ( p ∗ ( P t )) = [ P (1) − P ] = 0 bec ause P (1) = P .  Prop osition 4.3. Given a pr oje ct or P , m ∈ im(sf P ) if and only if ther e exists a pr oje ctor Q su ch that Q − P is c omp act, [ Q − P ] = m and is p ath-c onne cte d t o P . By the previo us theorem, im(sf P ) = im( δ P ). Ther efore, the prop osition follows from the definition of δ P . W e define the following prop erties: h1) P is path-connected to a pro jector Q s uc h that Q − P is compact and [ Q − P ] = m . h2) the image of p ∗ : π 1 ( P ( L ( E )) , P ) → π 1 ( P ( C ( E ) , p ( P )) is triv ial. Corollary 4.4. Given P ∈ P ( L ( E )) , we char acterise the kernel and the image of the sp e ct ra l flow sf P : (i) m ∈ im(sf P ) if and only if P fulfil ls pr op erty h1). (ii) im( p ∗ ) ∼ = ker( sf P ) . sf P is inje ctive if and only if P fulfil ls pr op erty h2). The is omorphism classes o f the kernel and the ima ge o f sf P depe nd o nly on the conjugacy class of P + L c ( E ) in P ( C ( E )). W e show that in many cases we ca n find a pro jector P such that sf P is an isomor phism. Lemma 4.5 . L et E b e a Banach sp ac e, and X, Y ⊂ E close d subsp ac es such that X ∼ = Y and X ⊕ Y = E . Then, the pr oje ct ors P X , P Y with r anges X and Y r esp e ctively, ar e c onne cte d by a c ontinuous p ath in P ( L ( E )) . A pro of of this can b e found in [ 22 , § 9 ] or in [ 19 ]. Prop osition 4 .6. L et X , Y ⊂ E b e as ab ove. Supp ose that X is isomorphic to its close d subsp ac es of c o-dimension m . Le t P b e the pr oje ct or onto X with kernel Y . Then P s at isfies the pr op erty h1) w.r.t. m . Pr o of. Let X m , R m ⊂ X b e clos e d subspa c es such that dim( R m ) = m and X m ∼ = X . W e hav e the following dec o mpositio ns and iso morphism: E = R m ⊕ X m ⊕ Y , X m ∼ = Y , R m ⊕ X m = X . By applying Lemma 4.5 to X m ⊕ Y and subspaces X m and Y , we obtain that P X m is connected to P Y . By applying it a seco nd time to E and subspaces X and Y , we obtain that P X is connected to P Y . Hence, P X is connected to P X m .  In Pr opo sition 4.6 , we required E to b e isomorphic to a cartesian pro duct of a space X with itself, but it suffices that E has a complemented subspace F fulfilling the requirements of Pr o pos ition 4.6 . In fact, if A t ∈ e H ( F ) is such that sf ( A t ) = m , then sf ( I ⊕ A t ) = m , by (iv) of Prop osition 3.3 . Prop osition 4.7. Given P ∈ P ( L ( E )) , t he map π : GL ( E ) → P ( L ( E )) , T 7→ T P T − 1 defines a princip al bund le with fib er GL ( X ) × GL ( Y ) , wher e X = Rang e( P ) and Y = ker( P ) . A pr o of of this can b e fo und in [ 10 , Theorem 2.1] or in [ 4 , P rop osition 1.2]. Bo th theorems a re stated in the more gene r al setting o f B anach a lgebras. SPECTRAL F LOW IN BANACH SP ACES 17 Corollary 4. 8. If GL ( E ) is simply-c onne cte d and GL ( X ) , GL ( Y ) ar e c onne ct e d, then sf P is inje ctive. Pr o of. Bec a use a lo cally trivial bundle is a Ser re fibration, we hav e a long exact sequence of homomorphisms that ends π 1 ( GL ( E ) , I ) π ∗ / / π 1 ( P ( L ( E )) , P ) ∆ / / π 0 ( GL ( X ) × GL ( Y ) , I ) . Thu s, if GL ( E ) is simply co nnected and GL ( X ) and GL ( Y ) are connected, the middle g roup is trivial, hence in ( 16 ) p ∗ is the triv ial map, thus δ P is injectiv e and sf P is injective.  Then, we hav e sufficient conditions for a Banach space to have at least o ne pro jector P such that sf P is an isomor phism. Theorem 4.9. L et E = X ⊕ X b e such that X is isomorphic to its hyp erplanes, GL ( E ) is simply-c onne cte d and GL ( X ) is c onne cte d. Then sf P X is an isomorphism. Pr o of. Tha t sf P X is sur jective follows from Pro positio n 4.6 . F ro m the cor ollary ab o ve, sf P X is also injective.  Let us consider the pa rticular cas e , where E is iso morphic to E × E and to its hyperplanes, and GL ( E ) is contractible to a p oint. This, in fact, is the cas e of the most common infinite-dimensional spaces as separ able Hilber t s paces, c 0 , ℓ p with p ≥ 1, a nd L p (Ω , µ ) with p > 1, C ( K, F ) for large classes of compac t spa ces K and Banach spaces F , and man y o thers. F or a richer lis t, see Theorem 2 of [ 28 ] and [ 17 , 6 , 20 , 19 ]. Sequence spaces ℓ p , ℓ ∞ and c 0 are a lso prime (see [ 5 , Theorem 2.2.4] and [ 18 ]), that is , they are isomo rphic to their complemented, infinite-dimensio nal subspaces. Th us, for ev ery pr o jector P such that Range( P ) and ker( P ) hav e infinite dimension, sf P is an isomor phism. T rivial sp e ctr al flow. When P is a pro jector with a finite-dimensio nal range or kernel, P + L c ( E ) is 0 or 1, then its co nnected comp onent in P ( C ( E )) is { 0 } or { 1 } . Hence, s f P = 0 . This is the case of finite- dimensional spaces. A space is said to be undeco mposa ble if the only pro jectors are as ab ov e. In [ 14 ], W. T. Gowers and B. Maurey showed the existence of an infinite-dimensional, undecomp osable space. Non-trivial and not surje ctive sp e ct r al flow. In [ 15 ], W. T. Gow er s and B. Maurey prov ed the existence o f a space isomorphic to their subspaces of co- dimension tw o, but not their h ype rplanes. If we denote by X a space with such a prop erty and by P the pro jector onto the first factor in E = X ⊕ X , then 2 ∈ im(sf P ) by Prop osition 4.6 . How ever, if 1 ∈ im(sf P ), by Pro p osition 4.3 and c2 ) in § 1.2 , X would be isomorphic to its hyperplanes . 4.1. The Douady space. W e show the exis tence o f a pro jector P such that δ P , and th us sf P , is not injective. Prop osition 4.10. L et E = X ⊕ X . Given T ∈ GL ( X ) , ther e exists a lo op x in the sp ac e of pr oje ctors such that ∆( x ) =  T 0 0 T − 1  , wher e ∆ is the c onne cting homomorphism in the se quenc e of Cor ol lary 4.8 18 GARRISI, D. Pr o of. Let M b e the op erator defined in the line ab ov e. Let U t ∈ GL ( E ) b e such that U (1) = M and U (0) = I . The existence of U follows from the fact that T ⊕ T ′ is connected to T T ′ ⊕ I ( see [ 19 ]). B e c ause M commutes with P X , the path P ( t ) = U ( t ) P X U ( t ) − 1 is a lo op in P ( L ( E )) with base po in t P X . W e denote its homo top y class by x . The path U t is a lifting path for P . Hence ∆( x ) = U (1) = M .  Let F and G b e Banach spaces s uc h that (i) every b ounded map G → F is compact; (ii) bo th F and G are isomo r phic to their hyper pla nes. The next Lemma follows from a more genera l r esult of A. Douady , [ 12 , Prop osi- tion 1]. W e briefly sketch the pro of by B. S. Mitjag in in [ 1 9 ]. Lemma 4.11. L et X = F ⊕ G , F and G as ab ove. Then, ther e exists a c ont inu ous, surje ctive homomorphism j : GL ( X ) → Z . Pr o of. Let T ∈ GL ( X ) b e an inv er tible op erator and S b e its inv erse. W e have  I F 0 0 I G  = T S =  T 11 S 11 + T 12 S 21 T 11 S 12 + T 12 S 22 T 21 S 11 + T 22 S 21 T 22 S 22 + T 21 S 12 .  A similar equality holds fo r S T . T aking the first element of the diagona ls of T S and S T , resp ectively , we have the following rela tions T 11 S 11 + T 12 S 21 = I F , S 11 T 11 + S 12 T 21 = I F . Because S 21 and T 21 are c ompact op e rators, T 11 and S 11 are the essential in verse o f each o ther . According to (e) of Pr opo sition 1.10 , T 11 is a F redholm op erator . W e define j ( T ) = ind ( T 11 ) . By (a) o f Prop osition 1.10 , there exists ε > 0 such that, if | | T ′ 11 − T 11 | | < ε , then T ′ 11 is a F redholm op erator and ind( T ′ 11 ) = ind( T 11 ). This prov es the co ntin uit y . Moreov er, given tw o inv ertible op erators T a nd S , we hav e j ( T S ) = ind( T S ) 11 = ind ( T 11 S 11 + T 12 S 21 ) = ind ( T 11 S 11 ) = ind( T 11 ) + ind( S 11 ) , where (b) and (c) of Prop osition 1.10 hav e b een used. Thus, j is a gro up homo- morphism. Le t F 1 and G 1 be hyperplanes of F a nd G , r espectively . W e define σ : F → F 1 , τ : G 1 → G F = h v i ⊕ F 1 , G = h w i ⊕ G 1 B : G → F, tw + y 7→ tv . where σ, τ ar e isomorphis ms, which exist b y (i). W e define T ( x, y ) = ( σ ( x ) + B ( y ) , τ ( P y )) where P : G → G is a pro jector onto G 1 . T is inv er tible and ind( T 11 ) = ind( σ ) = − 1. Because j is a homomorphis m, it is surjective.  Prop osition 4. 1 2. If E = X ⊕ X , wher e X is a dir e ct sum of two sp ac es F and G as in (i) and (ii) ab ove, then sf P X is surje ctive, but not inje ctive. SPECTRAL F LOW IN BANACH SP ACES 19 Pr o of. F rom the lemma a b ov e, for every k ∈ Z , there exis ts T k ∈ GL ( X ) such that j ( T k ) = k . By P rop osition 4.10 , there exists x k ∈ π 1 ( P ( L ( E )) , P X ) such that ∆( x k ) = T k ⊕ T − 1 k and x k 6 = x h if k 6 = h . Then π 1 ( P ( L ( E )) , P X ) has infinitely many elemen ts, while π 1 ( P c ( E ; P ) , P X ) is a finite group, by Prop osition 4.1 . Hence, in ( 1 5 ), i ∗ is not surjective, thus x k 6∈ im( i ∗ ) for infinitely many k ∈ Z . Hence, p ∗ ( x k ) 6 = 0 , δ P X ( p ∗ ( x k )) = 0 . bec ause the sequence ( 15 ) is exa ct; therefore , the kernel of δ P X is not trivia l. Be- cause E and X fulfill the hypothesis of P rop osition 4.6 , s f P X is surjective.  By [ 26 , Theorem 4.2 3], exa mples o f pair s of Banach spaces as in (i) and (ii) a re given b y ( ℓ p , ℓ 2 ), with p > 2. In the next pr o pos itio n, we show the existence of a pro jector P whos e ra nge and kernel are isomor phic to their hyperpla nes, but sf P = 0. Prop osition 4 .13. L et X = F ⊕ G , wher e F and G fulfil l pr op ert ies (i) and (ii). Then, sf P F = 0 . Pr o of. Let 0 ≥ − m ∈ im(sf P F ). Then, by (i) of Cor ollary 4.4 , P F is connected to a pro jector Q ∈ P ( L ( X )) such that Q − P F is compact and [ P F − Q ] = m . Let P m ∈ P ( L ( X )) b e a pro jector onto a subspa ce F m ⊂ F of co-dimensio n m such that P m ( I − P F ) = 0. Thus, P F − P m is compact and [ P F − P m ] = m . Therefore, Q − P m is compact and [ P m − Q ] = 0. By Pr opo sition 4.1 , Q is connected to P m . Hence, P F is c onnected to P m . By Theorem 1 .6 , there exists a co n tin uous path U t ∈ GL ( X ) such that U (0) = I , U (1) P F = P m U (1) . F rom these r elations, it follows that U (1) 11 ( F ) = F m , hence j ( U (1)) = − m , a nd j ( U (0 )) = 0. By the lemma a bov e, j ( U ( t )) is constant, therefor e m = 0.  5. The Fredhol m index of F A and the spectral flow Definition 5.1. A path A : R → L ( E ) of b ounded o pera tors is said to b e asymp- totically hyper bolic if the limits A (+ ∞ ) and A ( −∞ ) exis t and are hyperb olic op- erators . If A is also ess en tially hyperb olic, we ca n define the sp ectral flow as fo llo ws: Because the set of hyperb olic op erato r s is an op en subset o f L ( E ), there exists δ > 0 such that A ( t ) is hype r bolic for e very t ∈ ( −∞ , − δ ] ∪ [ δ, + ∞ ). W e set (17) sf ( A ) = s f ( A, [ − δ, δ ]) . The definition doe s not dep end on the choice of δ by (i,ii) of Prop osition 3.3 . Let P + ( A ) be the sp ectral pro jector P ( A ; { Re( z ) > 0 } ). Definition 5.2. An asymptotica lly hyperb olic pa th is called essential ly splitting if the following prop erties (i) P + ( A (+ ∞ )) − P + ( A ( −∞ )) is compact; (ii) [ A ( t ) , P + ( A (+ ∞ ))] is compact for every t ∈ R , hold, where [ A, P ] = AP − P A . The definition above co rresp onds to the o ne given in [ 2 , Theorem 6.3] when (i) holds. F o r shor t, we will r efer to essentially splitting as a path satisfying (i) and (ii). 20 GARRISI, D. Lemma 5. 3 . L et A b e an asymptotic al ly hyp erb olic and essential ly hyp erb olic p ath. It is also essential ly splitting if and only if P + ( A ( t )) − P + ( A ( s )) is c omp act for every t, s ∈ R . Pr o of. Supp ose A is essentially splitting. we denote by E + and E − the ranges of P + ( A (+ ∞ )) and P − ( A (+ ∞ )), resp ectively . With resp ect to the decomp osition E = E + ⊕ E − , we can write A ( t ) blo ck-wise: A ( t ) =  A + ( t ) K ± ( t ) K ∓ ( t ) A − ( t )  where K ± ( t ) and K ∓ ( t ) are compact op erators b y (ii). Then A ( t ) is a compa ct per turbation of A + ( t ) ⊕ A − ( t ). Ther efore (18) P + ( A ( t )) − P + ( A + ( t )) ⊕ P + ( A − ( t )) ∈ L c ( E ) for every t ∈ R . Because A (+ ∞ ) co mm utes with P + ( A (+ ∞ )) and P − ( A (+ ∞ )), K ± (+ ∞ ) , K ∓ (+ ∞ ) are nu ll op erator s, hence A (+ ∞ ) = A + (+ ∞ ) ⊕ A − (+ ∞ ). B y the definition o f essential s p ectrum and (b) of Theorem 1.10 , σ e ( A ( t )) = σ e ( A + ( t )) ∪ σ e ( A − ( t )). Therefore, P + ( A + (+ ∞ )) = I E + , A + ( t ) ∈ e H ( E + ) (19) P + ( A − (+ ∞ )) = 0 E − , A − ( t ) ∈ e H ( E − ) . (20) F rom the firs t part of ( 19 ) and Prop osition 2.11 , A + (+ ∞ ) ∈ X I E + ⊂ e H ( E + ) . where X I E + is the connected comp o nen t of I E + in e H ( E + ). Because e H ( E + ) is lo cally path-connected, X I E + is op en. Then, b y the seco nd part of ( 19 ) A + ( t ) ∈ X I E + for every t ∈ R . By Prop osition 2.11 , (21) P + ( A + ( t )) − I E + ∈ L c ( E ) for every t ∈ R . Similarly , from ( 20 ) and Pr opo sition 2.11 , we obtain (22) P + ( A − ( t )) ∈ L c ( E ) for every t ∈ R . By the definition of P + ( A (+ ∞ )), I E + ⊕ 0 E − = P + ( A (+ ∞ )). Thus, by ( 18 , 21 ) and ( 22 ), P + ( A ( t )) − P + ( A (+ ∞ )) ∈ L c ( E ) for every t ∈ R . Conv er sely , supp ose that each of the pr o jector s of the set { P + ( A ( t )) : t ∈ R } is a compact p erturbation of the others. Let a > 0 b e s uch that A ( s ) is hype r bolic for every | s | > a . B y the co n tin uit y o f P + , it follows tha t P + ( A (+ ∞ )) − P + ( A ( −∞ )) = lim s → + ∞ | s | >a  P + ( A ( s )) − P + ( A ( − s ))  . bec ause A is asymptotically hyperb olic. Hence, the left mem ber is the limit of a sequence of co mpact ope rators. Beca use L c ( E ) is a closed subset of L ( E ), we hav e prov ed (i). Prop erty (ii) follows from the equa lit y [ A ( t ) , P + ( A ( s ))] = [ A ( t ) , P + ( A ( t ))] + [ A ( t ) , P + ( A ( s )) − P + ( A ( t ))] , where the firs t summand of the r ig h t mem ber is zero and the second one is compact by hypothesis. In pa rticular, the equa lity holds for s > a , so we finish our pro of by taking the limit on the left member as s → + ∞ .  SPECTRAL F LOW IN BANACH SP ACES 21 5.1. The s p e ctral flo w and the F redholm index o f F A . Given a co n tin uous, bo unded path A t ∈ L ( E ), we denote b y F A the differential op erator F A : W 1 ,p ( R , E ) → L p ( R , E ) , F A ( u ) = du dt − A ( · ) u. Theorem 5. 4. L et A b e an asymptotic al ly hyp erb olic and essential ly s plitting p ath. Then F A is a F r e dholm op er ator and ind( F A ) = [ P − ( A ( −∞ )) − P − ( A (+ ∞ ))] A. Abb ondandolo and P . Ma jer prov ed it in [ 2 , Theorem 6.3 ] where E is a Hilb ert space a nd p = 2. How ever, the theor em, like muc h of the con ten t o f their w ork, can be genera lis ed to Banach spaces as in [ 13 , Theorem 3.3]. Theorem 5 . 5. L et A b e an asymptotic al ly hyp erb olic, essential ly splitting and es- sential ly hyp erb olic p ath. Then, sf ( A ) = − [ P − ( A ( −∞ )) − P − ( A (+ ∞ ))] Pr o of. Let δ > 0 as in ( 17 ). B y Lemma 5.3 , the co nstan t path P ≡ P + ( A ( δ )) is an s-section for P + ( A t ) on [ − δ, δ ] in the sense of Definition 3.4 . Hence, sf ( A ) = [ P + ( A ( δ )) − P + ( A ( − δ ))] . Because A is hyperb olic on ( −∞ − δ ] ∪ [ δ, + ∞ ), the path P + ( A t ) is co n tin uous on this s ubset. By (iii) of Theor em 1.12 , [ P − ( A ( −∞ )) − P − ( A (+ ∞ ))] = − [ P + ( A ( δ )) − P + ( A ( − δ ))] = − sf ( A ) .  F rom Theor e m 5.5 and Theorem 5.4 we have the final result Theorem 5.6 . If A is essential ly hyp erb olic, essential ly splitting and asymptotic al ly hyp erb olic, then (23) ind( F A ) = − sf ( A ) . Let A ( t ) = A 0 ( t ) + K ( t ), where A 0 ( t ) is hyperb olic a nd A 0 , A a r e asymptotically hyperb olic. K ( t ) is compa ct and A 0 ( t ) E − ⊆ E − , A 0 ( t ) E + ⊆ E + ; E − = E − ( A 0 ( ±∞ )) , E + = E + ( A 0 ( ±∞ )) . The second line tells us that P + ( A 0 (+ ∞ )) = P + ( A 0 ( −∞ )) and thus P + ( A (+ ∞ )) − P + ( A ( −∞ )) is co mpact. F rom the first line, it follows that [ A ( t ) , P + ( A (+ ∞ ))] is compact. Th us, by Theo rem 5.6 , we confirm the guess of A. Abbondando lo and P . Ma jer in [ 2 , § 7], that for paths satis fying the hypotheses o f [ 2 , Theorem E], corres p onding to those listed ab ov e, the rela tion ( 23 ) holds. 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