Spectrum for compact operators on Banach spaces

Submitted to Journal of the Mathematical Society of Japan Spectrum for compact operators on Banach spaces By Luis Barreira, Davor Dragičević Claudia Valls (Received Nov. 22, 203) (Revised Jan. 23, 204) Abstract. For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix the set of its eigenvalues. We give a characterization of all possible spectra explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.. Introduction We study a notion of spectrum for a nonautonomous dynamics inspired on a corresponding notion introduced by Sacker Sell in [9] in a finite-dimensional space. This can be seen as a generalization of the spectrum of a matrix (the set of its eigenvalues). More precisely, let (A m ) m Z be a sequence of invertible d d matrices consider the dynamics x m+ = A m x m, m Z. For a constant sequence A m = A, a number a R is of the form a = log µ for some eigenvalue µ of A if only if the constant sequence (e a A) m Z admits an exponential dichotomy. For an arbitrary sequence of matrices, an appropriate notion of spectrum was introduction by Aulbach Siegmund in [] (following work in [] in the case of continuous time) by declaring that a is in the spectrum if the sequence (e a A m ) m Z admits an exponential dichotomy. For related work we refer the reader to [2, 3, 8, 0, 2]. We consider a corresponding notion in an infinite-dimensional setting. Namely, given a sequence (A m ) m Z of compact linear operators acting on a Banach space, we define its spectrum as the set Σ of all a R such that the sequence (e a A m ) m Z does not admit an exponential dichotomy. We emphasize that in the notion of an exponential dichotomy we only assume the dynamics to be invertible along the unstable direction. Our main result describes all possible types of spectra how they relate to certain natural invariant subspaces (see Theorem 3). In particular, we show that each trajectory has lower upper Lyapunov exponents inside the same connected component of the spectrum. We note that this mimics the behavior in the multiplicative ergodic theorem in the particular case of Lyapunov regular trajectories, which is the typical behavior for example in all mechanical systems on a compact energy hypersurface. 200 Mathematics Subject Classification. Primary 37D99. Key Words Phrases. Compact operators, Exponential dichotomies, Spectra.

2 L. Barreira, D. Dragičević C. Valls Moreover, we provide examples of sequences for which the spectrum takes a form that does not occur in finite-dimensional spaces. More precisely, we provide explicit examples of sequences of compact linear operators acting on the Hilbert space of L 2 functions on the circle with the induced Lebesgue measure for which the spectrum is of the form for some numbers Σ = [a j, b j ] or Σ = [a j, b j ] (, b ], j= j= b a > b 2 a 2 > b 3 a 3 >, respectively, with lim n + a n = or lim n + a n = b >. Finally, we consider briefly the case of a one-sided sequence of compact linear operators we describe corresponding notions results. 2. Spectrum on the line 2.. Exponential dichotomies Let (A n ) n Z be a sequence of compact linear operators acting on a Banach space X = (X, ). For each m n, let () A(m, n) = Am A n, m > n, Id, m = n. We say that (A n ) n Z admits an exponential dichotomy if:. there exist projections P n : X X for n Z satisfying (2) A n P n = P n+ A n for n Z such that each map (3) A n Ker P n : Ker P n Ker P n+ is invertible; 2. there exist constants D, λ > 0 such that (4) A(m, n)p n De λ(m n) for m n (5) A(m, n)q n De λ(n m) for m n, where Q n = Id P n (6) A(m, n) = (A(n, m) Ker P m ) : Ker P n Ker P m for m < n.

Spectrum for compact operators on Banach spaces 3 The sets Im P n Im Q n are called, respectively, the stable unstable spaces of the exponential dichotomy. The following result shows in particular that they are uniquely determined. Proposition. For each n Z, } Im P n = v X : sup A(m, n)v < + m n Im Q n consists of all v X for which there exists a sequence (x m ) m n X such that x n = v, x m = A m x m for m n sup m n x m < +. Moreover, dim Im Q n < + for n Z. Proof. By (4) we have (7) sup A(m, n)v < + m n for v Im P n. Conversely, if (7) holds for some v X, then it follows from (4) that (8) sup A(m, n)q n v < +. m n By (5), for m n we have that is, Whenever Q n v 0, we obtain Q n v De λ(m n) A(m, n)q n v, D eλ(m n) Q n v A(m, n)q n v. sup A(m, n)q n v = +, m n which contradicts to (8). Hence, Q n v = 0 so v Im P n. Now take v Im Q n consider the sequence x m = A(m, n)v for m n. Clearly, x n = v x m = A m x m for m n. Moreover, it follows from (5) that sup m n x m < +. Conversely, one can show that there is no v X \ Im Q n for which there exists a sequence (x m ) m n X with the properties in the proposition. Indeed, it follows from (2) (4) that P n v = A(n, m)p m x m De λ(n m) x m for m n. Hence, if P n v 0, then sup m n x m = +. It remains to show that the unstable spaces are finite-dimensional. Let B n = v Im Q n : v } take n N such that e λn > D. We claim that B n A(n, 0)B 0. For each v B n,

4 L. Barreira, D. Dragičević C. Valls there exists x Im Q 0 such that v = A(n, 0)x. If x >, then it follows from (5) that < D eλn x A(n, 0)x = v, which contradicts to the assumption that v. Hence x so B n A(n, 0)B 0. Since B 0 is bounded A(n, 0) is compact, the set A(n, 0)B 0 is relatively compact so B n is compact. This shows that Im Q n is finite-dimensional. On the other h, it follows from (2) that the dimensions dim Im Q m are independent of m Z. This completes the proof of the proposition. 2.2. Spectrum The spectrum of a sequence (A n ) n Z of compact linear operators is the set Σ of all numbers a R such that the sequence (e a A n ) n Z does not admit an exponential dichotomy. For each a R n Z, let ( S a (n) = v X : sup e a(m n) A(m, n)v ) } < + m n let U a (n) be the set of all vectors v X for which there exists a sequence (x m ) m n X such that x n = v, x m = A m x m for m n We note that if a < b, then ( sup e a(m n) x m ) < +. m n S a (n) S b (n) U b (n) U a (n) for n Z. By Proposition, if a R \ Σ, then X = S a (n) U a (n) for n Z the projections P n Q n associated to the sequence (e a A n ) n Z satisfy Im P n = S a (n) Im Q n = U a (n). Moreover, by (2), for each a R \ Σ, the numbers dim S a (n) dim U a (n) are independent of n. We shall simply denote them by dim S a dim U a. Proposition 2. The following statements hold:. The set Σ is closed. Moreover, for each a R \ Σ we have S a (n) = S b (n) U a (n) = U b (n) for all n Z all b in some open neighborhood of a. 2. Take a, a 2 R \ Σ with a < a 2. Then [a, a 2 ] Σ if only if dim U a > dim U a2. 3. For each c / Σ, the set Σ [c, + ) consists of finitely many closed intervals. Proof. Given a R \ Σ, there exist projections P n for n Z satisfying (2) a

Spectrum for compact operators on Banach spaces 5 constant λ > 0 such that for m n for m n. Therefore, for each b R, for m n e a(m n) A(m, n)p n De λ(m n) e a(m n) A(m, n)q n De λ(n m) e b(m n) A(m, n)p n De (λ a+b)(m n) e b(m n) A(m, n)q n De (λ+a b)(n m) for m n. Therefore, b R \ Σ whenever a b < λ it follows from Proposition that S b (n) = S a (n) U b (n) = U a (n) for n Z. For statement 2, assume that [a, a 2 ] Σ. If dim U a = dim U a2, then U a (n) = U a2 (n) S a (n) = S a2 (n) for n Z. By Proposition, there exist projections P n for n Z constants λ, λ 2 > 0 such that for i =, 2 we have (9) e ai(m n) A(m, n)p n D i e λi(m n) for m n (0) e ai(m n) A(m, n)q n D i e λi(n m) for m n. For each a [a, a 2 ], by (9), e a(m n) A(m, n)p n D e λ(m n) for m n similarly, by (0), e a(m n) A(m, n)q n D 2 e λ2(n m) for m n. Taking λ = minλ, λ 2 } D = maxd, D 2 }, we conclude that [a, a 2 ] R \ Σ, which contradicts to the initial assumption. Therefore, dim U a = dim U a2. For the converse, assume that dim U a > dim U a2 let b = inf a R \ Σ : dim U a = dim U a2 }. Since dim U a > dim U a2, it follows from statement that a < b < a 2. We claim that b Σ. Otherwise, either dim U b = dim U a2 or dim U b dim U a2. In the first case, by statement there exists ε > 0 such that dim U b = dim U a2 b R\Σ for b (b ε, b]. This contradicts to the definition of b. In the second case, by statement there exists ε > 0 such that dim U b dim U a2 b R\Σ for b [b, b+ε). Again this contradicts to the definition of b. Hence, b Σ [a, a 2 ] Σ. For statement 3, we proceed by contradiction. Write dim U c = d assume that Σ [c, + ) contains at least d+2 disjoint closed intervals I i = [α i, β i ], for i =,..., d+2,

6 L. Barreira, D. Dragičević C. Valls where α β < α 2 β 2 < < α d+2 β d+2 +. For i =,..., d +, take c i (β i, α i+ ). It follows from statement 2 that d > dim U c > dim U c2 > > dim U cd+, which is impossible. This completes the proof of the proposition. 3. Structure of the spectrum Our main result describes all possible forms of the spectrum for a sequence of compact linear operators acting on a Banach space. Theorem 3. Let (A n ) n Z be a sequence of compact linear operators for which the spectrum is neither nor R.. One of the following alternatives holds: (a) Σ = I k n=2 [a n, b n ], where I = [a, b ] or I = [a, + ), for some numbers () b a > b 2 a 2 > > b k a k ; (b) Σ = I k n=2 [a n, b n ] (, b k ], where I = [a, b ] or I = [a, + ), for some numbers a n b n as in (); (c) Σ = I n=2 [a n, b n ], where I = [a, b ] or I = [a, + ), for some numbers (2) b a > b 2 a 2 > b 3 a 3 > with lim n + a n = ; (d) Σ = I n=2 [a n, b n ] (, b ], where I = [a, b ] or I = [a, + ), for some numbers a n b n as in (2) with b := lim n + a n >. 2. Let (c k ) k be a finite or infinite sequence of numbers such that c k (b k+, a k ) for each k. For each n Z v S ck (n) U ck+ (n) \ 0}, we have [ lim inf m + log A(m, n)v, lim sup m m + ] log A(m, n)v [a k+, b k+ ]. m Moreover, there exists a sequence (x m ) m n X such that x n = v, x m = A m x m for m n [ ] lim inf m m log x m, lim sup m m log x m [a k+, b k+ ]. Proof. We first show that Σ has one of the forms in alternatives (a) (d). Assume first that Σ is not given by alternatives (a) (b) take c / Σ. By statement 3 of Proposition 2, the set Σ (c, + ) consists of finitely many disjoint closed intervals I,..., I k. We note that Σ (, c ), since otherwise we would have Σ = I I k,

Spectrum for compact operators on Banach spaces 7 which contradicts to our assumption. Now we observe that there exists c 2 < c such that c 2 / Σ (c 2, c ) Σ. Indeed, otherwise we would have (, c ) Σ = (, a] for some a < c thus, Σ = (, a] I I k, which again contradicts to our assumption. Proceeding inductively, we obtain a decreasing sequence (c n ) n N such that c n / Σ (c n+, c n ) Σ for each n N. Now there are two possibilities: either lim n + c n = or lim n + c n = b for some b R. In the first case, it follows from statement 2 of Proposition 2 that Σ is given by alternative (c). In the second case, it follows from statement 3 of Proposition 2 that (a, + ) Σ = I [a n, b n ], where I = [a, b ] or I = [a, + ), for some sequences (a n ) n N (b n ) n N as in (2) with b = lim n + a n. Again by statement 3 of Proposition 2, we have (, b ] Σ so Σ is given by alternative (d). This concludes the proof of statement. For statement 2, we first note that it follows easily from statement 2 of Proposition 2 that the subspaces n=2 E k (n) = S ck (n) U ck+ (n) are independent of the choice of numbers c k. Since c k / Σ, the sequence (e c k A n ) n Z admits an exponential dichotomy so there exist projections P n for n Z satisfying (2) (3) constants λ, D > 0 such that (3) A(m, n)p n De (c k λ)(m n) for m n A(m, n)q n De (λ+c k)(n m) for m n, where Q n = Id P n. By Proposition, we have Im P n = S ck (n) for n Z. Hence, v E i (n) belongs to Im P n it follows from (3) that Letting c k b k+, we obtain lim sup m + m log A(m, n)v c k λ < c k. lim sup m + m log A(m, n)v b k+. Similarly, since c k+ / Σ, the sequence (e c k+ A n ) n Z admits an exponential dichotomy so there exist projections P n for n Z satisfying (2) (3) constants µ, D > 0 such that A(m, n)p n De (c k+ µ)(m n) for m n

8 L. Barreira, D. Dragičević C. Valls (4) A(m, n)q n De (µ+c k+)(n m) for m n, where Q n = Id P n. By Proposition, we have Im Q n = U ck+ (n) for n Z. Hence, v E i (n) belongs to Im Q n it follows from (4) that Thus, letting c k+ a k+, v De (µ+c k+)(m n) A(m, n)v for m n. lim inf m + m log A(m, n)v µ + c k+ > c k+ lim inf m + m log A(m, n)v a k+. The last statement in the theorem can be proved similarly. 4. Examples Now we provide explicit examples of sequences (A n ) n Z for which the spectrum Σ is given by the last two alternatives in Theorem 3 (the other alternatives already occur in finite-dimensional spaces). Let S = λ C : λ = } consider the space X = L 2 (S ) with respect to the induced Lebesgue measure on S. We recall that X is a Banach space when equipped with the norm induced by the scalar product x, y = x(z)y(z) dz, S already identifying functions that are equal Lebesgue-almost everywhere. Moreover, for each K L 2 (S S ), the operator A: X X defined by (5) (Af)(z) = K(z, w)f(w) dw S is compact (see [4]). For example, one can take K(z, w) = k(z/w), with k X. We also consider the orthonormal basis e n : n Z} of X, where e n (z) = z n, for z S n Z. Each function k X can be written in the form k = n Z k ne n for some numbers k n R. Moreover, one can easily verify that Ae n = 2πk n e n for each n Z, taking K(z, w) = k(z/w). Example. Take numbers a n b n as in (2) with lim n + a n =. We consider the linear operators A, B : X X such that Ae i = e bi e i Be i = e ai e i for i 0, writing e 0 = e 0, e 2n = e n e 2n = e n for n N. Both A B are of

Spectrum for compact operators on Banach spaces 9 the form in (5) taking, respectively, k(z) = 2π eb0 + 2π n= ( e b2n z n + e ) b2n z n k(z) = 2π ea0 + 2π n= ( e a2n z n + e ) a2n z n. Now we consider the sequence of compact linear operators A, n 0, A n = B, n < 0. For each a R i 0, we have e a(m n) A(m, n)e i = C i (m, n)e i, where e (bi a)(m n), m, n 0, (6) C i (m, n) = e (bi a)m (ai a)n, m 0, n < 0, e (ai a)(m n), m, n < 0. Take a > b. For each x X, we have A(m, n)x 2 = C i (m, n) 2 x, e i 2. Since a j b j b, we have C i (m, n) e (b a)(m n) thus, A(m, n)x 2 e 2(b a)(m n) This shows that i=0 i=0 x, e i 2 = e 2(b a)(m n) x 2. A(m, n)x e (a b)(m n) x for m n so (e a A n ) n Z admits an exponential dichotomy with projections P n = Id. Now take a (b j, a j ) for some j 2. Let P n be the projection given by P n e i = 0 for i < j P n e i = e i for i j. For each x X m n, we have thus, A(m, n)p n x 2 = i=j C i (m, n) 2 x, e i 2 e 2(bj a)(m n) x 2 A(m, n)p n x e (a bj)(m n) x.

0 L. Barreira, D. Dragičević C. Valls Similarly, for each x X m n, we have thus, j 2 A(m, n)q n x 2 = C i (m, n) 2 x, e i 2 e 2(aj a)(m n) x 2 i=0 A(m, n)q n x e (aj a)(n m) x. This shows that the sequence (e a A n ) n Z admits an exponential dichotomy. Therefore, Σ j= [a j, b j ]. In order to show that Σ = j= [a j, b j ], assume that [a j, b j ] \ Σ is nonempty for some j N take a [a j, b j ] \ Σ. Then the sequence (e a A n ) n Z admits an exponential dichotomy say with projections P n. Let Y be the subspace of X generated by e i let B n be the restriction of A n to Y. Clearly, the sequence (e a B n ) n Z admits an exponential dichotomy, say with projections P n. Moreover, either P n = Id or P n = 0, but both alternatives are impossible in view of (6). Example 2. Now take numbers a n b n as in (2) with lim n + a n = b >. We consider the sequence of compact linear operators (A n ) n Z such that A n e 0 = e b, n 0, e n2, n < 0 A n e i = e b i /( + i2 n ), n 0, e ai /( + i2 n ), n < 0, i. For each a R i 0, we have e a(m n) A(m, n)e i = C i (m, n)e i, where e (b a)(m n), m, n 0, C 0 (m, n) = e (b a)m+an n2, m 0, n < 0, e am+an n2 +m 2, m, n < 0 e (bi a)(m n) / m k=n ( + i2 k ), m, n 0, C i (m, n) = e (bi a)m (ai a)n / m k=n ( + i2 k ), m 0, n < 0, e (ai a)(m n) / m k=n ( + i2 k ), m, n < 0 for i. Notice that + j2 n. One can proceed as in the previous example to show that for a > b, the sequence (e a A n ) n Z admits an exponential dichotomy with projections P n = Id. Now take a (b j, a j ). For i j m n, we have C i (m, n) e (bj a)(m n). Similarly, for i < j m n, we have C i (m, n) e (aj a)(m n) max i<j p= ( + i2 p ).

Spectrum for compact operators on Banach spaces Proceeding as in the previous example, one can show that the sequence (e a A n ) n Z admits an exponential dichotomy, with projections P n given by P n e i = 0 for i < j P n e i = e i otherwise. By statement 3 of Proposition 2, we also have (, b ] Σ. Hence, Σ [a j, b j ] (, b ]. j= Moreover, one can proceed as in the previous example to show that in fact Σ = [a j, b j ] (, b ]. j= 5. Spectrum on the half-line 5.. Preliminaries Let (A n ) n N be a sequence of compact linear operators acting on a Banach space X. For each m n we define A(m, n) as in (). We say that (A n ) n N admits an exponential dichotomy if there exist projections P n for n N satisfying (2) (3) there exist constants D, λ > 0 such that (4) (6) hold with m, n N. The following result can be obtained repeating the first part of the proof of Proposition. It shows that the images of the projections P m are uniquely determined. Proposition 4. For each n N, we have } Im P n = v X : sup A(m, n)v < + m n. On the other h, the images of the projections Q n are not uniquely determined, in contrast to what happens in the line. Proposition 5. Assume that the sequence (A m ) m N admits an exponential dichotomy with respect to projections P m. Moreover, let P m, for m N, be projections such that P ma(m, n) = A(m, n)p n for m, n N A n Ker P n : Ker P n Ker P n+ is invertible. Then (A m ) m N admits an exponential dichotomy with respect to the projections P m if only if Im P n = Im P n for n N. Proof. If (A m ) m N admits an exponential dichotomy with respect to the projections P m, then it follows from Proposition 4 that } Im P n = v X : sup A(m, n)v < + m n = Im P n.

2 L. Barreira, D. Dragičević C. Valls Now assume that Im P n = Im P n. Then In particular, P n P n = P n P np n = P n. it follows from (4) (5) that P n P n = P n (P n P n) = (P n P n)q n A(n, )(P P )v = A(n, )P (P P )v for m, n N v X. Since we obtain De λ(n ) (P P )v = e λ De λn (P P )Q v e λ De λn P P Q v = e λ De λn P P A(, m)a(m, )Q v = e λ De λn P P A(, m)q m A(m, )v e 2λ D 2 e λn λm P P A(m, )v A(n, m)(p m P m) = A(n, m)(p m P m)q m = (P n P n)a(n, m)q m = (P n P n)a(n, )Q A(, m)q m = A(n, )(P P )Q A(, m)q m = A(n, )(P P )A(, m)q m, A(n, m)p mv A(n, m)p m v + A(n, m)(p m P m)v = A(n, m)p m v + A(n, )(P P )A(, m)q m v De λ(n m) v + e 2λ D 3 e λ(n m) P P v = D e λ(n m) v for n m some constant D > 0. Similarly, letting Q m = Id P m we obtain A(n, m)q mv A(n, m)q m v + A(n, m)(p m P m)v = A(n, m)q m v + A(n, )(P P )A(, m)q m v De λ(m n) v + e 2λ D 3 e λ(m n) P P v = D e λ(m n) v for n m. This shows that the sequence (A m ) m N admits an exponential dichotomy with respect to the projections P m. The following proposition is a crucial step of this part.

Spectrum for compact operators on Banach spaces 3 Proposition 6. Assume that the sequence (A m ) m N admits an exponential dichotomy with respect to projections P m P m. Then Proof. By Proposition 5, we have dim Im Q m = dim Im Q m < +, m N. X = Im P n Im Q n = Im P n Im Q n all complements of Im P n have the same dimension. In view this proposition we will denote by d a the unstable dimension of the sequence (e a A m ) m N when it admits an exponential dichotomy. 5.2. Spectrum Given a sequence (A m ) m N of compact linear operators, its spectrum is the set Σ of all numbers a R such that the sequence (e a A m ) m N does not admit an exponential dichotomy. For each a R n N, let ( S a (n) = v X : sup e a(m n) A(m, n)v ) } < +. m n Clearly, S a (n) is a linear space if a < b, then S a (n) S b (n). The following result can be obtained repeating the proof of Proposition 2 with dim U ai replaced by d ai. Proposition 7. The following statements hold:. The set Σ is closed. Moreover, for each a R \ Σ we have S a (n) = S b (n) for all n N all b in some open neighborhood of a. 2. Take a, a 2 R \ Σ with a < a 2. Then [a, a 2 ] Σ if only if d a > d a2. 3. For each c / Σ, the set Σ [c, + ) consists of finitely many closed intervals. The following is our main theorem for the half-line. It describes all possible forms of the spectrum for a sequence of compact linear operators. Theorem 8. Let (A n ) n N be a sequence of compact linear operators for which the spectrum is neither nor R.. Property of Theorem 3 holds. 2. Let (c k ) k be a finite or infinite sequence of numbers such that c k (b k+, a k ) for each k. For each n N v S ck (n), we have lim sup m + m log A(m, n)v b k+. Proof. The proof of statement is the same as in the proof of Theorem 3. For the second statement, since c k / Σ, the sequence (e c k A n ) n N admits an exponential dichotomy

4 L. Barreira, D. Dragičević C. Valls so there exist projections P n for n N satisfying (2) (3) constants λ, D > 0 such that (7) A(m, n)p n De (c k λ)(m n) for m n A(m, n)q n De (λ+c k)(n m) for m n, where Q n = Id P n. By Proposition 7, the space S ck (n) is independent of the choice of c k Im P n = S ck (n) for n N. Hence, for v S ck (n), it follows from (7) that letting c k b k+, lim sup m + m log A(m, n)v c k λ < c k lim sup m + m log A(m, n)v b k+. This completes the proof of the theorem. 6. The case of continuous time In this section we consider briefly the case of continuous time we describe corresponding results to those in the former sections. Let X be a Banach space. We recall that a family T (t, s) for t, s R with t s of bounded linear operators acting on X is called an evolution family if:. T (t, t) = Id for t R; 2. T (t, s) = T (t, r)t (r, s) for t r s. We say that T (t, s) admits an exponential dichotomy if:. there exist projections P (t): X X for t R satisfying for t s such that each map is invertible; 2. there exist constants D, λ > 0 such that P (t)t (t, s) = T (t, s)p (s) T (t, s) Ker P (s): Ker P (s) Ker P (t) T (t, s)p (s) De λ(t s) for t s T (t, s)q(s) De λ(s t) for t s,

Spectrum for compact operators on Banach spaces 5 where Q(s) = Id P (s) for t < s. T (t, s) = (T (s, t) Ker P (t)) : Ker P (s) Ker P (t) The spectrum of an evolution family T (t, s) is the set Σ of all numbers a R such that the evolution family T a (t, s) = e a(t s) T (t, s) does not admit an exponential dichotomy. One can easily adapt the arguments in the proof of Theorem 3 to establish the following result. Theorem 9. Let T (t, s) be an evolution family for which the spectrum is neither nor R. If there exists ρ 0 such that T (t, s) is compact whenever t > s + ρ, then property of Theorem 3 holds. The compactness assumption holds in particular for the evolution families associated to some linear delay equations with ρ > 0 (see [5]) some linear parabolic partial differential equations with ρ = 0 (see [6]). It is also straightforward to establish a version of Theorem 9 for evolution families on the half-line. Now we consider the particular class of evolution families with bounded growth we show how one can use directly the results for discrete time to obtain corresponding results for this class. More precisely, we obtain the particular case of Theorem 9 for this class with a very simple proof applying Theorem 3, instead of having the need to adapt all arguments in its proof. Theorem 0. K, c > 0 such that Let T (t, s) be an evolution family with the property that there exist (8) T (t, s) Ke c(t s) for t s. If there exists ρ 0 such that T (t, s) is compact whenever t > s + ρ, then one of the following alternatives hold:. Σ = ; 2. Σ = k n= [a n, b n ], for some numbers a n b n as in (); 3. Σ = k n= [a n, b n ] (, b k ], for some numbers a n b n as in (); 4. Σ = n= [a n, b n ], for some numbers a n b n as in (2) with lim n + a n = ; 5. Σ = n= [a n, b n ] (, b ], for some numbers a n b n as in (2) with b := lim n + a n >. Proof. We first recall that for an evolution family satisfying (8) the following properties are equivalent (see [7, Theorem.3]):

6 L. Barreira, D. Dragičević C. Valls. T (t, s) admits an exponential dichotomy; 2. taking α > ρ, the sequence (A n ) n Z with admits an exponential dichotomy. A n = T (αn, α(n )), for n Z, Hence, Σ coincides with the spectrum of the sequence (A n ) n Z. Moreover, in view of the compactness assumption for T (t, s) since α > ρ, each operator A n is compact. The desired conclusion follows now readily from Theorem 3 since it follows from (8) that Σ (, c]. Acknowledgements. L.B. C.V. were supported by FCT/Portugal through UID/MAT/04459/203. D.D. was supported in part by an Australian Research Council Discovery Project DP500007, Croatian Science Foundation under the project IP-204-09-2285 by the University of Rijeka research grant 3.4..2.02. References [] B. Aulbach S. Siegmund, A spectral theory for nonautonomous difference equations, in New Trends in Difference Equations (Temuco, 2000), Taylor & Francis, London, 2002, pp. 45 55. [2] C. Chicone Yu. Latushkin, Evolution Semigroups in Dynamical Systems Differential Equations, Mathematical Surveys Monographs, 70, American Mathematical Society, Providence, RI, 999. [3] S.-N. Chow H. Leiva, Dynamical spectrum for time dependent linear systems in Banach spaces, Japan J. Indust. Appl. Math., (994), 379 45. [4] J. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 990. [5] J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer- Verlag, New York-Heidelberg, 977. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 98. [7] D. Henry, Exponential dichotomies, the shadowing lemma homoclinic orbits in Banach spaces, Resenhas, (994), 38 40. [8] R. Johnson, K. Palmer G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 8 (987), 33. [9] R. Sacker G. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (978), 320 358. [0] R. Sacker G. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 3 (994), 7 67. [] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 4 (2002), 243 258. [2] G. Wang Y. Cao, Dynamical spectrum in rom dynamical systems, J. Dynam. Differential Equations 26 (204), 20. Luis Barreira Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 049-00 Lisboa, Portugal E-mail: barreira@math.tecnico.ulisboa.pt Davor Dragičević School of Mathematics Statistics, University of New South Wales, Sydney NSW 2052, Australia E-mail: ddragicevic@math.uniri.hr

Spectrum for compact operators on Banach spaces 7 Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 049-00 Lisboa, Portugal E-mail: cvalls@math.tecnico.ulisboa.pt