EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON LOCALLY COMPACT SPACES WITH BANACH FIBERS

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1 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON LOCALLY COMPACT SPACES WITH BANACH FIBERS Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Abstract. We study evolutionary semigroups generated by a strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from nonautonomuous abstract Cauchy problems and C 0 -semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: roughness of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green s function for a linear skew-product flow, pointwise dichotomy versus global dichotomy, and evolutionary semigroups along trajectories of the flow. 1. Introduction The spectral theory of linear skew-product flows (or processes) with finite dimensional fibers is, by now, a well-developed area in asymptotics theory of differential equations (see, e.g., [14, 15, 20, 38, 46, 47, 49, 50]). J. Hale in [14, p.60] stressed that this theory should be extended to the infinite dimensional setting. Indeed, in recent years significant progress has been made in the study of linear skew-product flows (LSPFs) with Banach fibers ([8, 9, 10, 28, 48]) over a compact 1991 Mathematics Subject Classification. 47D06, 34G10, 34C35. Key words and phrases. Evolutionary semigroups, spectral mapping theorems, linear skew-product flows, exponential dichotomy. Supported by the National Science Foundation under the grant DMS and by the Summer Research Fellowship of the University of Missouri. Supported by the National Science Foundation under the grant DMS Supported by the University of Missouri Research Board. 1

2 2 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH metric space (see also [6, 7, 26]). An impressive list of possible applications can be found in [8, 14, 15, 16, 48]. In this paper we consider strongly continuous linear skew-product semiflows on bundles with Banach fibers over a locally compact metric space. Our philosophy is not to start with a pointwise construction of stable and unstable foliations (as in [50, 8, 9, 28, 46, 48]), but instead to begin by associating with the linear skew-product flow an evolutionary semigroup of operators, {T t } t 0, on the space of continuous sections of the bundle. We prove two facts concerning the spectrum of this semigroup that relate its spectral properties to the asymptotic properties of the linear skew-product flow. The first of these facts is the Spectral Mapping Theorem (Theorem 3.3) for {T t } t 0 and its generator Γ, e tσ(γ) = σ(t t )\{0}, t > 0, while the second concerns the description of the spectral projections (Theorem 4.1). As a result of this approach, we not only answer questions concerning LSPFs, but also address the theory of evolutionary semigroups associated with strongly continuous evolutionary families as studied in the theory of nonautonomous Cauchy problems and the general theory of C 0 semigroups (see [12, 13, 18, 27, 33, 36, 42]). Consequently, classical theorems on the exponential dichotomy of differential equations (see, e.g., [11, 31]) can easily be extended to the case concerning differential equations with unbounded coefficients. Some of the results appearing here were announced in [23]. The connection between evolutionary semigroups and the dichotomy of LSPFs is not new. In one of the first papers in this direction, J. Mather [30] proved that the LSPF generated by the differential of a diffeomorphism of a smooth manifold is Anosov (i.e., is hyperbolic or, in the terminology of the present paper, exponentially dichotomic) if and only if the corresponding evolutionary operator is hyperbolic (it s spectrum does not intersect the unit circle). This led to the notion of the Mather spectrum M := σ(t ). Here T = T 1 and σ( ) denotes the usual spectrum of an operator. This notion is widely used in the theory of hyperbolic dynamical systems (see [40] for further references). The articles [4, 19] proved the spectral mapping theorem for evolutionary semigroups in the finite dimensional setting, and they described the

3 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 3 spectral subbundles of these operators via the spectral subbundles of the corresponding LSPF. Moreover, they related the spectrum of Γ to the dynamical spectrum, Σ, of the LSPF, as defined by R. Sacker and G. Sell in [47]. In the present paper we proceed in the opposite direction: we derive the existence of spectral subbundles from the existence, and description, of the Riesz projections corresponding to T ; the properties of Σ are described via the spectrum of the generator, σ(γ). These techniques build upon those used in [1] which addresses the finite dimensional setting and uses some C -algebraic methods. These methods were also used in [25] to prove the results for a uniformly continuous cocycle on a Hilbert space. The results have recently been generalized in [43] to the case involving a strongly continuous cocycle on a compact metric space with Banach fibers. The important articles [43, 44, 45] linked this theory with the theory of evolutionary semigroups associated with strongly continuous evolutionary families (as initiated and developed in [12, 13, 18, 27, 36]). As we will see below, the latter situation corresponds to a LSPF over the translations of R; the evolutionary family can be thought of as the propagator of a well-posed differential equation on a Banach space [33, 34, 51]. For this situation, in the Banach space setting, the Spectral Mapping Theorem has been proven in [22, 23] (see [17],[33] and [39] for a general discussion on this theorem) and the Spectral Projection Theorem in [22, 23, 24, 42, 44, 45]. The Spectral Mapping Theorem appearing in the present paper applies to any evolutionary semigroup associated with a LSPF over a nonperiodic flow on a locally compact space, Θ, but it also provides a new proof for semigroups arising from evolutionary families, i.e., when Θ = R and the flow is translation on R. In fact, the same idea applies to evolutionary semigroups in the space of divergence-free sections of a bundle a situation which is important for applications to hydromagnetodynamics [5]. Our proof of the Spectral Projection Theorem for LSPFs also differs from that in [43]. Here, we extend the algebraic technique used in [1, 25] (for Hilbert spaces) so that it applies in the general setting of Banach spaces. One advantage to this approach is that we are able to discuss the case of discrete time t Z and pointwise dichotomies (see Section 5; cf. [8]).

4 4 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH We now make the previous discussion more explicit by establishing some notation and outlining the main results. For t R, let ϕ t be a continuous flow on a locally compact metric space Θ. Throughout the paper, the aperiodic trajectories of the flow are assumed to be dense in Θ. Let X be a Banach space and let L s (X) be the space of bounded linear operators on X endowed strong-operator topology. Let Φ : Θ R + L s (X) be a continuous (semi)cocycle over ϕ t. Since Θ is locally compact, we will allow Φ to grow at infinity not faster then exponentially. The linear skew-product (semi)flow ˆϕ t is defined by the following formula (see Section 2) ˆϕ t : Θ X Θ X, (θ, x) (ϕ t θ, Φ(θ, t)x). Example 1.1. As a very particular example, which will be referred as the norm continuous compact setting, assume Θ is compact and A: Θ L(X) is a continuous operator-valued function. Then, for each θ Θ, the operator Φ(θ, t) can be thought of as a solving operator for the variational equation dx dt = A(ϕt θ)x(t), θ Θ, t R. (1.1) Here, Φ is uniformly continuous and takes invertible values in L(X). Example 1.2. As another example (see also [10]), which will be referred to as the norm continuous line setting, assume Θ = R, and let A: R L(X) be a bounded, continuous operator-valued function. Let {U(τ, s)} τ s be the propagator of the nonautonomous differential equation dx = A(t)x(t), t R; (1.2) dt that is, the solution x( ) of (1.2) satisfies x(τ) = U(τ, s)x(s). This can be related to the previous example by defining Φ(θ, t) = U(θ + t, θ) and identifying the flow as translation, ϕ t θ = θ + t for θ Θ = R and t R. Since the propagator U satisfies [11] the identity U(τ, s) = U(τ, r)u(r, s), for τ r s, and U(s, s) = I, Φ is a cocycle over ϕ t. These two examples show (see also [10]) how both the variational equation (1.1) and the nonautonomous equation (1.2) can be addressed in terms of linear skew-product flows on a locally compact metric space. We stress that the strongly continuous setting considered in the present paper (as opposed to the norm continuous situation described above)

5 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 5 allows one to study the equations (1.1) and (1.2) even in the situation where the operators given by A( ) are unbounded. Given a LSPF ˆϕ t, we associate to it a semigroup {T t } t 0 of evolutionary operators defined on C 0 (Θ, X), the space of strongly continuous functions vanishing at infinity (with the sup-norm), by (T t f)(θ) = Φ(ϕ t θ, t)f(ϕ t θ), θ Θ, f C 0 (Θ, X). The generator of such a semigroup will be denoted by Γ. For the line setting this semigroup becomes a well-known [12, 13, 18, 27, 34, 36, 44, 45] evolutionary semigroup: (T t f)(τ) = U(τ, τ t)f(τ t), τ R, f C 0 (R, X). The generator, Γ, of the evolutionary semigroups arising in the norm continuous setting of examples 1.1 and 1.2 can be expressed, respectively, as (Γf)(θ) = d dt f ϕt (θ) +A(θ)f(θ), and (Γf)(τ) = df t=0 dτ +A(τ)f(τ). A discussion on the delicate question about when these formulas hold for the strongly continuous setting can be found in [34, 35]. If, for the line setting, A 0 A(τ) generates a C 0 semigroup {e ta 0 } t 0, then (T t f)(τ) = e ta 0 f(τ t) and Γ is the closure of d/dt + A 0. If Φ is the differential of a diffeomorphism on a smooth manifold Θ, then T t is the push-forward operator and Γ is the Lie derivative. In Section 3, for the strongly continuous locally compact setting, we prove the following theorem. Spectral Mapping Theorem. The spectrum σ(γ) is invariant with respect to translations along the imaginary axis, and the spectrum σ(t t ), t > 0, is invariant with respect to rotations centered at origin. Moreover, σ(t t )\{0} = e tσ(γ). The proof of this theorem develops some ideas from [30] on localization of almost-eigenfunctions for evolutionary operators. In Section 4 we discuss exponentially dichotomic LSPFs. The terminology exponential dichotomy, as in [8] (see also [15, 16, 28, 48]), refers to the existence of a strongly continuous, bounded projectionvalued function P : Θ L s (X). This P ( ) defines a ˆϕ t -invariant splitting Θ X = X P + X Q with uniform exponential decay of Φ P on X P

6 6 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH and of Φ 1 Q on X Q, Q := I P. The subscripts here denote the restrictions of the LSPF; since we do not assume that Φ is invertible, the existence of Φ 1 Q is part of the definition of exponential dichotomy. The main result in Section 4 is the following theorem where an operator T is called hyperbolic if σ(t ) T =. If T is hyperbolic, its Riesz projection corresponding to σ(t ) D is denoted by P. (Here, T and D denote the unit circle and unit disk in C.) Spectral Projection Theorem. Assume T is hyperbolic. Then its Riesz projection, P, is an operator of multiplication. That is, if f C 0 (Θ, X), then (Pf)(θ) = P (θ)f(θ) for some strongly continuous bounded projection-valued function P ( ): Θ L s (X). We use this theorem to show that the LSPF ˆϕ t has exponential dichotomy with P ( ) if and only if T is hyperbolic, and we relate P ( ) with the Riesz projection for T, as above. The related result for compact Θ was proved in [43] (see also [42, 44, 45] for Θ = R), where a quite different method is used. We stress that our method also works for discrete time t Z, that is, when ϕ is a homeomorphism, and Φ : Θ Z L s (X). For continuous time, t R, together with the Spectral Mapping Theorem, this shows that the LSPF ˆϕ t has exponential dichotomy with P ( ) if and only if Γ is invertible. This generalizes some results in [3, 37]. In particular (see [22, 23]), for Θ = R and A(τ) A 0 this shows that the growth bound for any C 0 semigroup {e ta 0 } t 0 on X coincides with the spectral bound for d/dt + A 0 on C 0 (R, X). A similar fact (see [22]) for L p -spaces can be used in a variety of contexts (cf. [32, 52]). Our method of proof exploits the invertibility properties of a particular Banach algebra of weighted translation operators on C 0 (Θ, X). To prove the Spectral Projection Theorem, we consider for each θ Θ a weighted shift operator, π θ (T ), acting on the space l (Z, X) and given by the diagonal matrix: π θ (T ) = diag{φ(ϕ n 1 θ, 1)} n Z S; here, S denotes the shift operator on l (Z, X): (x n ) n Z (x n 1 ) n Z. The hyperbolicity of π θ (T ) is equivalent to the existence of the discrete exponential dichotomy for ˆϕ t along the trajectory through θ. These operators were, in fact, introduced in [16] for the line setting, and in

7 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 7 [1, 8, 25] for the compact setting. A byproduct of our approach is the following theorem. Pointwise Dichotomy Theorem. The LSPF ˆϕ t has exponential dichotomy on Θ if and only if π θ (T ) is hyperbolic for every θ Θ, and sup{ (λ π θ (T )) 1 : λ T, θ Θ} <. The theorems listed above help to unify a number of ideas concerning spectral properties of dynamical systems. Recall that the dynamical spectrum, Σ, for the LSPF ˆϕ t (see [47]) is the set of all λ R such that the LSPF, ˆϕ t λ, corresponding to the cocycle e λt Φ(θ, t) does not have exponential dichotomy. The Bohl spectrum, B, for equation (1.1) is the set of λ R such that the equation x = [A(τ) λ]x does not have exponential dichotomy (see [11]). Using the above theorems, one can relate all of these sets (in the general strongly continuous, locally compact setting) as follows: Σ = B = ln M = σ(γ) R = ln σ(t ) \ {0}. (1.3) Finally, consider an autonomous equation x (t) = A 0 x(t), t R, where A 0 generates a C 0 semigroup, {e ta 0 } t 0. Assume, in addition, that the spectral mapping theorem is valid for {e ta 0 } t 0. This happens, e.g., if A 0 is a sectorial operator (see [33] for a detailed discussion). Then each of the sets in (1.3) coincides with {z + iξ : z σ(a 0 ), ξ R} R. If X is a Hilbert space and A 0 is self-adjoint, then the last set is just σ(a 0 ). This shows that for abstract Cauchy problems for nonautonomous differential equations and for LSPFs in Banach spaces, the spectrum of Γ should play the same role as the spectrum of A 0 for autonomous equations. In Section 5 we give several consequences of the results mentioned above. The list of these topics includes: roughness of the dichotomy, dichotomy and solutions of nonhomogenious equations (1.1) (1.2), Green s function for a LSPF, pointwise dichotomy versus global dichotomy, and evolutionary semigroups along trajectories of the flow. The paper concludes by listing several open problems. The following notation is used throughout the paper: Θ denotes a locally compact metric space; for θ Θ and δ > 0, B(θ, δ) denotes an open ball of radius δ centered at θ. For any Banach space X, L(X) denotes the space of bounded linear operators on X, and L s (X) denotes

8 8 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH this space endowed with the strong-operator topology. For any operator A on X, its domain is denoted by D(A) and its spectrum by σ(a). In C, let T = {z : z = 1}, and D = {z : z 1}. For a set of operators d (n), n Z, in L(X), diag{d (n) } n Z acts on l (Z, X) as the infinite matrix consisting of diagonal entries d (n). For a projection P (or P, or a projection-valued function P ( )), we use the letter Q (respectively, Q, Q( )) to denote the complementary projection: Q = I P. 2. Evolutionary Semigroups: Preliminaries 2.1. Evolutionary semigroups. Let ϕ t be a continuous flow on Θ, and let Φ: Θ R + L(X) be a strongly continuous, exponentially bounded cocycle over ϕ t. That is, we assume the function (θ, t) Φ(θ, t)x to be continuous from Θ R + to X for each x X, and there exist C, ω > 0 such that for every θ Θ, a) Φ(θ, t + s) = Φ(ϕ t θ, s)φ(θ, t) for t, s > 0; b) Φ(θ, 0) = I; c) Φ(θ, t) L(X) Ce ωt for all t R +. Note that the operators Φ(θ, t) are not assumed to be invertible. The last inequality holds automatically if Θ is compact. The linear skew-product flow (LSPF) associated with Φ is the map ˆϕ t : Θ X Θ X, ˆϕ t (θ, x) = (ϕ t θ, Φ(θ, t)x), t 0. (2.1) We note that for exponentially bounded cocycles, the map ˆϕ t is continuous if and only if the corresponding cocycle is strongly continuous. To a cocycle Φ over a flow ϕ t, one can associate the family of operators {T t } t 0 in B(C 0 (Θ, X)) defined by (T t f)(θ) = Φ(ϕ t θ, t)f(ϕ t θ), θ Θ, f C 0 (Θ, X). (2.2) It s easy to check that this defines a semigroup of operators: T t+s = T t T s, T 0 = I. As noted below, this evolutionary semigroup is a strongly continuous semigroup on C 0 (Θ, X) if and only if Φ is a strongly continuous, exponentially bounded cocycle. The generator will be denoted by Γ.

9 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 9 Theorem 2.1. {T t } t 0, as defined in (2.2), is a C 0 semigroup on C 0 (Θ, X) if and only if Φ is an exponentially bounded, strongly continuous cocycle. Proof. If Φ is a strongly continuous cocycle with Φ(θ, t) Ce ωt for some C, ω > 0, then {T t } t 0 can be seen to be strongly continuous as in [25] or [43]. Conversely, assume {T t } t 0 is a C 0 semigroup. Clearly, Φ is exponentially bounded. Fix θ 0 Θ, t 0 R, and x X. Let ɛ > 0. For θ Θ, and t R, consider Φ(θ, t)x Φ(θ 0, t 0 )x Φ(θ 0, t)x Φ(θ 0, t)x + Φ(θ 0, t)x Φ(ϕ t t 0 θ 0, t 0 )x + Φ(ϕ t t 0 θ 0, t 0 )x Φ(θ 0, t 0 )x. (2.3) Let D be a compact set in Θ containing θ 0 in its interior. Choose α: Θ [0, 1] with compact support such that α(θ) = 1 for all θ D. Define f C 0 (Θ, X) as f(θ) = α(θ)x, and note that for θ D and t R, Φ(θ, t)x = (T t f)(ϕ t θ). First consider the middle term on the right-hand side of (2.3) and choose δ 1 > 0 such that ϕ t t 0 θ 0 D for all t t 0 < δ 1. Choose δ 2 (0, δ 1 ] such that t t 0 < δ 2 implies T t f T t 0 f ɛ/3. Then for t t 0 < δ 2, Φ(θ 0, t)x Φ(ϕ t t 0 θ 0, t 0 )x = (T t f)(ϕ t θ 0 ) (T t 0 f)(ϕ t 0 (ϕ t t 0 θ 0 )) = (T t f)(ϕ t θ 0 ) (T t 0 f)(ϕ t θ 0 ) sup T t f(θ) T t 0 f(θ) < ɛ/3. θ Θ Secondly, since T t f is continuous, there exists δ > 0 such that d(θ 1, θ 2 ) < δ implies T t 0 f(θ 1 ) T t 0 f(θ 2 ) < ɛ/3. Since t ϕ t θ 0 is continuous, there exists δ 3 (0, δ 2 ] such that t t 0 < δ 3 implies d(ϕ t θ 0, ϕ t 0 θ 0 ) < δ. Then, t t 0 < δ 3 implies Φ(ϕ t t 0 θ 0, t 0 )x Φ(θ 0, t 0 )x = T t 0 f(ϕ t 0 (ϕ t t 0 θ 0 )) T t 0 f(ϕ t 0 θ 0 ) = T t 0 f(ϕ t θ 0 ) T t 0 f(ϕ t 0 θ 0 ) < ɛ/3. Finally, choose δ > 0 so that B(θ 0, δ ) D, and θ B(θ 0, δ ) implies (T t f)(ϕ t θ) (T t f)(ϕ t θ 0 ) < ɛ/3.

10 10 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH We note that δ = δ (t) depends on t, but on the compact interval [t 0 δ 3, t 0 + δ 3 ], the map t T t f(ϕ t ) is uniformly continuous, so δ may be chosen independent of t. Therefore, if t t 0 < δ 3 and θ B(θ 0, δ ), then (2.3) shows that Φ(θ, t)x Φ(θ 0, t 0 )x < ɛ Examples. We give several examples of cocycles and corresponding evolutionary semigroups. Example 2.2. Norm continuous compact setting. Let ϕ t be a continuous flow on a compact metric space Θ, and let A: Θ L(X) be (norm) continuous. For each θ Θ consider the equation dx dt = A(ϕt θ)x(t), θ Θ, t R. (2.4) Let Φ(θ, t), t R, be the solving operator for (2.4): x(t) = Φ(θ, t)x(0). Then Φ is a cocycle and A(θ) = d dt t=0 Φ(θ, t). Denote (df)(θ) = d dt f ϕt (θ). The generator Γ of the group (2.2) is given as follows: t=0 (Γf)(θ) = (df)(θ) + A(θ)f(θ), (2.5) and its domain D(Γ) is, in this case, D(Γ) = {f C 0 (Θ, X) : df C 0 (Θ, X)}. Equations of the type (2.4) arise from two sources. Firstly, they can be thought of as a linearization of a nonlinear equation in X in the vicinity of a compact invariant set Θ for the nonlinear equation. Secondly, Θ might be a compact hull, Θ = closure{a( + τ) : τ R}, of a given function a: R L(X) (see [14, 46]). Example 2.3. Strongly continuous compact setting. We now allow the operators A(θ) in (2.4) to be unbounded. A strongly continuous (semi)cocycle Φ is said to solve (2.4) if: For every θ Θ and every x θ D(A(θ)) the function t x(t) := Φ(θ, t)x θ is differentiable for t > 0, x(t) D(A(ϕ t θ)), and x( ) satisfies (2.4). By Theorem 2.1 the cocycle generates an evolutionary semigroup given by (2.2). This setting might occur if, after the linearization of a nonlinear equation in X, the Frechet derivative A(θ) is an unbounded operator. Numerous examples of this strongly continuous setting can be found in [8].

11 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 11 Example 2.4. To be more specific, in the setting of Example 2.3, assume A(θ) A 0, θ Θ, where A 0 generates a C 0 semigroup {e ta 0 } t 0 on X. The cocycle Φ for (2.4) is Φ(θ, t) = e ta 0. In the tensor product C 0 (Θ, X) = C 0 (Θ, R) X one can express the evolutionary semigroup defined in (2.2) by T t = V t e ta 0, where V t f := f ϕ t. Then (see, e.g., [33, p. 23]) the generator Γ 0 of {T t } t 0 is the closure of the operator Γ 0f = df + A 0 f with D(Γ 0) = {f C 0 (Θ, X) : df C 0 (Θ, X), f : Θ D(A 0 ), df A 0 f C 0 (Θ, X)}. Example 2.5. Suppose, in the setting of Example 2.3, that A(θ) = A 0 + A 1 (θ), θ Θ, (2.6) where A 0 is a generator of a C 0 semigroup {e ta 0 } t 0 on X and A 1 : Θ L s (X) is (strongly) continuous and bounded on Θ. Note that A 1 ( ) defines an operator A 1 B(C 0 (Θ, X)) by the rule (A 1 f)(θ) = A 1 (θ)f(θ). For Γ 0 as in Example 2.4 we have: Proposition 2.6. Assume that there exists a strongly continuous cocycle Φ that solves (2.4) with A( ) as in (2.6). Then the generator Γ of the evolutionary semigroup (2.2) is given by (Γf)(θ) = (df)(θ) + A 0 f(θ) + A 1 (θ)f(θ), D(Γ) = D(Γ 0 ). (2.7) Proof. Since Φ defines a classical solution of (2.4) (2.6), Φ also is a mild solution of (2.4), that is, Φ satisfies Φ(θ, t)x = e ta 0 x + t 0 e (t τ)a 0 A 1 (ϕ τ θ)φ(θ, τ)x dτ, (2.8) for x X, θ Θ, t 0. (We point out the interesting Theorem 5.1 in [10] where the existence of a mild solution Φ was proved for any equations (2.4) (2.6).) Since Γ 0 generates a C 0 semigroup, and A 1 B(C 0 (Θ, X)), the operator Γ = Γ 0 +A 1 also generates (see, e.g., [39, p. 77]) a C 0 semigroup, {S t } t 0, which is the unique solution of the integral equation S t f = e tγ 0 f + t 0 e (t τ)γ 0 A 1 S τ f dτ, f C 0 (Θ, X), t 0. (2.9)

12 12 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH We show that S t = T t. Indeed, for f C 0 (Θ, X) define g = t Then (2.8) implies g(θ) = t 0 e (t τ)a 0 A 1 (ϕ τ (ϕ t θ))φ(ϕ t θ, τ)f(ϕ t θ) dτ = Φ(ϕ t θ, t)f(ϕ t θ) e ta 0 f(ϕ t θ) = (T t f)(θ) ( e tγ 0 f ) (θ). 0 e (t τ)γ 0 A 1 T τ f dτ. Therefore, {T t } t 0 satisfies (2.9), and so T t = S t. The generator, Γ = Γ 0 + A 1, of this semigroup is given by (2.7). Example 2.7. Norm continuous line setting. Let Θ = R, and assume A: R L(X) is a bounded continuous function. Let U(τ, s), τ, s R, denote the propagator for the equation dx = A(t)x(t), t R. (2.10) dt This means that the solution x( ) of (2.10) satisfies x(τ) = U(τ, s)x(s). Denote ϕ t (τ) = τ + t and Φ(τ, t) = U(τ + t, τ), for τ, t R (cf. [10]). Then Φ is a cocycle. In this case, the group (2.2) is given by (T t f)(τ) = U(τ, τ t)f(τ t), f C 0 (R, X), and the generator Γ is given by the formula: (Γf)(τ) = df dτ +A(τ)f(τ), D(Γ) = {f C 0(R, X) : f C 0 (R, X)}. Example 2.8. Strongly continuous line setting. In the case of Θ = R, consider a well-posed differential equation (2.10) with, generally, unbounded operators A(τ). Let {U(t, s)} t s be the associated strongly continuous evolutionary family of operators in L(X). This means that (τ, s) U(τ, s)x is continuous for each x X, U(τ, s) = U(τ, r)u(r, s) for τ r s, U(s, s) = I, and U(τ, s) Ce ω(τ s) for some C, ω > 0. The flow ϕ t, the cocycle Φ, and the evolutionary semigroup can be defined exactly as in the previous example. If, in particular, A(τ) A 0 is a generator of a C 0 semigroup {e ta 0 } t 0 on X, then U(τ, s) = e (τ s)a 0, and Γ is the closure of Γ = d dt +A 0, D(Γ ) = {f C 0 (R, X) : f +A 0 f C 0 (R, X), f : R D(A 0 )}. For Θ = R consider now the case where each of the operators A(t), t R in (2.10) is an (unbounded) operator on X with dense domain.

13 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 13 For each t R, denote the domain of A(t) by D t := D(A(t)). Consider the following nonautonomous abstract Cauchy problem (cf. [34]): dx dt = A(t)x(t) for t s R, and x(s) = x s, (2.11) where x s D s, s R. We say (see [34]) that the evolutionary family {U(τ, s)} τ s solves (2.11) if x( ) = U(, s)x s is a unique solution of (2.11) for every x s D s, that is x( ) is differentiable, x(t) D t for t s, and (2.11) holds. Proposition 2.9. Assume that a strongly continuous evolutionary family {U(τ, s)} τ s solves (2.11). Then the generator Γ of the evolutionary semigroup (T t f)(τ) = U(τ, τ t)f(τ t) on C 0 (R, X) is the closure of the operator Γ, with domain consisting of differentiable functions f such that f(t) D t and Γ f C 0 (R, X), defined as follows: (Γ f)(τ) = df dτ + A(τ)f(τ), τ R. (2.12) Proof. Fix s R and x s D s. For any smooth α : R R with compact support supp α [s, ), consider a function f C 0 (R, X), defined by f(τ) = α(τ)u(τ, s)x s for τ > s and f(τ) = 0 for τ s. Then Γf = Γ f. Indeed (2.13) (T t f)(τ) = α(τ t)u(τ, τ t)u(τ t, s)x s = α(τ t)u(τ, s)x s for τ t > s, and zero otherwise. Hence, d dt (T t f)(τ) = α (τ)u(τ, s)x s, τ R. t=0 On the other hand, since τ U(τ, s)x s satisfies (2.11), one has: df dτ = α (τ)u(τ, s)x s +α(τ) d dτ U(τ, s)x s = α (τ)u(τ, s)x s +α(τ)a(τ)u(τ, s)x s. To finish the proof, we need to show that linear combinations of functions f, as in (2.13), are dense in C 0 (R, X). To see this, first observe that the set of finite sums β j v j with arbitrary v j X and smooth β j : R R with compact support, is dense in C 0 (R, X).

14 14 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Now, consider any function g = βv with fixed v X and smooth β : R R with compact support. We show that g can be approximated by a sum of functions f as in (2.13). To see that, fix ɛ > 0. For every τ 0 supp β, the map (τ, s) U(τ, s)v, τ s is continuous at the point (τ 0, τ 0 ) by the definition of {U(τ, s)}. Hence, there exist s 0 s 0 such that for the interval I 0 = I(τ 0 ) := (s 0, s 0), containing τ 0, one has U(τ, s 0 )v v ɛ/2 for τ I 0. Thus, we obtain an open covering of supp β, formed by I(τ 0 ), τ 0 supp β. Take a finite subcovering {I j } n j=1. For I j := (s j, s j), one has U(τ, s j )v v ɛ/2 for τ I j, j = 1,..., n. Consider a smooth partition of unity {γ j } n j=1 for {I j } n j=1, that is, smooth functions γ j : R [0, 1] such that n γ j (τ) = 1 for τ supp β, and supp γ j I j, j = 1,..., n. j=1 Since the D sj are dense in X, one can choose v j D sj such that v v j ɛ/2. Define the function n h(τ) := β(τ)γ j (τ)u(τ, s j )v j, τ R. j=1 Since supp γ j (s j, s j), the function h is a sum of functions f as in (2.13). Also, g(τ) h(τ) = n β(τ) n γ j (τ)v β(τ)γ j (τ)u(τ, s j )v j j=1 j=1 n n β(τ) γ j (τ) v v j + γ j (τ) v j U(τ, s j )v j j=1 ɛ max β(τ), τ and the proof is completed An algebra of weighted translation operators. In Section 4 we study the spectrum of T = T 1, that is, the invertibility of b = λi T. This operator belongs to an algebra, B, of weighted translation operators which we now define. Let C b (Θ; L s (X)) denote the set of strongly continuous and bounded functions a: Θ L s (X). For a C b (Θ, L s (X)), set a u := sup θ Θ a(θ) L(X). j=1

15 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 15 Such a function induces a multiplication operator on C 0 (Θ, X) defined by (M a f)(θ) = a(θ)f(θ). The mapping a M a is an isometry from the Banach space C b (Θ, L s (X)) to B(C 0 (Θ, X)) and so the operator M a will be denoted simply by a, its norm given by a = a B(C0 (Θ,X)) = a u. Let A denote the set of all such multiplication operators a = M a B(C 0 (Θ, X)). Now let ϕ = ϕ 1 be a homeomorphism on Θ. Denote by V the translation operator on C 0 (Θ, X) given by (V f)(θ) = f(ϕ 1 θ). Define B to be the set of all operators b B(C 0 (Θ, X)) such that b = a k V k, where a k A, and b 1 := a k <. (2.14) Proposition If the set of aperiodic points of ϕ is dense in Θ, then the representation (2.14) of an element b B is unique. Proof. This follows from the observation that for any polynomial b N = Nk= N a k V k in B(C 0 (Θ, X)), where a k A and N N, b N B(C0 (Θ,X)) a k B(C0 (Θ,X)), k N. (2.15) To prove (2.15), first note that by replacing b by bv k it suffices to consider only the case k = 0. Fix ɛ > 0. For a A, a B(C0 (Θ,X)) = sup sup θ Θ so there exists a nonperiodic point θ 0 of {ϕ t } t R, and a x X, x = 1, such that a 0 (θ 0 )x X a 0 B(C0 (Θ,X)) ɛ. x X =1 Choose δ > 0 such that for B = B(θ 0, δ), ϕ k (B) ϕ j (B) = for k j, k, j N. Choose a continuous function α : Θ [0, 1] such that α(θ 0 ) = 1 and α(θ) = 0 for θ / B. Define f C 0 (Θ, X) by α(ϕ k θ)x, if θ ϕ k (B), k N f(θ) = 0 otherwise. Then f C0 (Θ,X) = 1 and N b B(C0 (Θ,X)) bf C0 (Θ,X) = max a k (θ)α(ϕ k θ)x θ Θ k= N X = max max a k(θ)α(ϕ k θ)x X a 0 (θ 0 )x X θ Θ k N a 0 B(C0 (Θ,X)) ɛ. a(θ)x X,

16 16 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Proposition (B, 1 ) is a Banach algebra. Proof. The norm 1 is an algebra norm. To see that it is complete, consider a 1 -Cauchy sequence {b (n) } n=1 in B, b (n) = a (n) k V k. Set a k = 1 - lim m a(m) k, k Z. For every m, n in Z, a (m) k a (n) k b(m) b (n) 1. So a sequence ( a (m) k ) k Z l 1 (Z, R) converges to ( a k ) k Z in l 1 (Z, R). Hence b = k a k V k is an element of B. For every ɛ > 0 and for sufficiently large m, n, the inequality ɛ b (m) b (n) 1 = a (m) k a (n) k N k= N a (m) k a (n) k holds for every N N. Taking the limit as n, and then the limit as N shows b (m) b 1 0. In Section 4, we consider the relationship between the invertibility of a weighted translation operator b in B and a family of representations (weighted shift operators), π θ (b), in B(l (Z, X)). To define the representation π θ of B, denote by S the shift operator on l (Z, X): S : (x n ) n Z (x n 1 ) n Z, (x n ) n Z l (Z, X). For a A and θ Θ, let π θ (a) = diag{a(ϕ n θ)} n Z. Defining π θ (av ) = π θ (a)s gives a continuous homomorphism from B into B(l (Z, X)): for b = k a k S k in B, π θ (b) = π θ (a k )S k. (2.16) With this in mind, let C denote the set of operators in B(l (Z, X)) of the form d = diag{d (n) } n Z, where d (n) L(X), and consider the Banach algebra (D, 1 ) of operators on l (Z, X) of the form d = d k S k where d k C, and d 1 := d k B(l (Z,X)) <. The following lemma points out, in particular, that D is an inverseclosed subalgebra of B(l (Z, X)). The proof follows exactly as in Lemma 1.6 and Remark 1.7 of [24].

17 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 17 Proposition Let d(θ) = assume that d k (θ)s k be an operator in D, and sup d k (θ) B(l (Z,X)) <. θ Θ If d(θ) 1 B(l (Z, X))) for each θ Θ and if for some C > 0, sup d(θ) 1 B(l (Z,X)) < C, θ Θ then d(θ) 1 D for each θ Θ. Moreover, if d(θ) 1 = then sup c k (θ) B(l (Z,X)) <. θ Θ 3. Spectral Mapping Theorem In this section we prove that the spectral mapping property c k (θ)s k, e tσ(γ) = σ(t t )\{0}, t > 0, (3.1) holds for any evolutionary semigroup (2.2) with generator Γ. As usual, the nonperiodic points of {ϕ t } t R are assumed to be dense in Θ. For a general C 0 semigroup {T t } t 0 generated by an operator Γ, the spectral inclusion e tσ(γ) σ(t t )\{0} is well known, as are examples showing that the inclusion is, in general, proper. Moreover, equality holds for the point spectrum and residual spectrum [33], so to prove (3.1) we focus on the approximate point spectrum, σ ap ( ). By a standard rescaling technique [33], it suffices to consider only T = T 1. We build on the ideas of [30] and observe, in particular, that the ɛ-eigenfunctions of T obey the following localization principle: if 1 σ ap (T ), then for every N N, there exists a point θ 0 Θ, a function f C 0 (Θ, X) with f = 1, and an open neighborhood D of θ 0 such that supp f 2N j=0ϕ j (D) and T f f = O( 1 ). Using f, we N construct a function g C 0 (Θ, X) such that Γg = O( 1 ) g, hence N showing that 0 σ ap (Γ). Before beginning the proof of (3.1), we prove two technical lemmas which verify the existence of an appropriate f C 0 (Θ, X) and aid in the subsequent construction of g, as described above.

18 18 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Lemma 3.1. Let A L(X) and assume 1 σ ap (A). For each N 2, there exists x X, x = 1, such that a) A N x x X 1 8 ; b) A k x X 2 for k = 0, 1,..., 2N. Proof. Set c = 2N k=0 A k. From the identity A k I = (A k A + I)(A I), for k = 2,..., 2N, it follows that for any x X, (A k I)x c (A I)x, k = 0, 1,..., 2N. (3.2) Since 1 σ ap (A), there exists a x X, x = 1, such that (A I)x 1. Set k = N in (3.2) to obtain a). Moreover, (3.2) shows that for 8c k = 0, 1,..., 2N, A k x (A k I)x + x c (A I)x + 1 c 1 8c Now let θ 0 Θ be a nonperiodic point of ϕ t, and fix s (0, 1) and N 2. If B is a neighborhood of θ 0, then for θ Θ we use the notation R(θ) = {t R : t N, ϕ t x B}. Lebesgue measure on R will be denoted by m. Lemma 3.2. There exists a sufficiently small open neighborhood, B, of θ 0 and a continuous bump function α: Θ [0, 1] such that: a) α(ϕ t θ 0 ) = 1, for t s 4 ; b) α(ϕ t θ 0 ) = 0, for s t 2N; c) m(r(θ)) 2s, for all θ Θ. For the important case Θ = R with ϕ t θ = θ+t, this is trivial: for any θ 0 R take B = (θ 0 s, θ 0 + s), and α : R [0, 1] with supp α B and α(θ) = 1 for x (θ 0 s/4, θ 0 + s/4). Proof. We begin with: Claim 1. For sufficiently small ɛ > 0, and all ɛ ɛ, if δ < ɛ, then θ B := B(θ 0, δ) implies ϕ t θ / B provided s t 5N. 2 Proof of Claim 1. Suppose, to the contrary, that there exists a sequence δ n 0 such that for some θ n B n := B(θ 0, δ n ) and some t n with s t 2 n 5N, one has ϕ tn θ n B n. Passing to a subsequence, one can assume t n t for some t with s t 2 5N. Since the map (θ, t) ϕ t θ from Θ R to Θ is continuous at the point (θ 0, t ), and since θ n B n, it follows that θ n θ 0. Thus t n t implies

19 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 19 ϕ tn θ n ϕ t θ 0. On the other hand, ϕ tn θ n B n implies ϕ tn θ n θ 0, as t n t. But then ϕ t θ 0 = θ 0, which contradicts the assumption that θ 0 is a nonperiodic point of ϕ t. This proves Claim 1. Now let δ < ɛ, and set B = B(θ 0, δ) as in Claim 1. Choose an open neighborhood B of θ 0 such that B B. Denote: B := ϕ t (B ), C := ϕ t (B ). (3.3) t s 4 t s 4 Since C B, there exists a continuous α: Θ [0, 1] such that α(θ) = 1 for θ C, and α(θ) = 0 for θ / B. Note that since θ 0 B, it follows that {ϕ t θ 0 : t s } C and so 4 α(ϕ t θ 0 ) = 1 whenever t s. This proves a). 4 To prove b) we first prove Claim 2. If θ B and if s t 2N, then ϕ t θ / B. Proof of Claim 2. Suppose, to the contrary, that there exists a θ B and a t R satisfying s t 2N and ϕ t θ B. By the definition (3.3) of B, there exist θ 1 and θ 2 in B such that θ = ϕ t 1 θ 1 and ϕ t θ = ϕ t 2 θ 2, for some numbers t 1, t 2 with t 1, t 2 s. This implies 4 ϕ t +t 1 t 2 θ 1 B. (3.4) Note that and t + t 1 t 2 2N + s 4 + s 4 5N t + t 1 t 2 t t 1 t 2 s s 4 s 4 = s 2, and so (3.4) contradicts Claim 1. This proves Claim 2. Since, in particular, θ 0 B and α(θ) = 0 for θ / B, Claim 2 proves part b) of the lemma. To prove c), fix θ 1 Θ and denote the orbit by O(θ 1 ) = {ϕ t θ 1 : t N}. Clearly, m(r(θ 1 )) = 0 provided O(θ 1 ) B =, so consider the case O(θ 1 ) B. Fix any θ O(θ 1 ) B. Then θ = ϕ t 1 θ 1 for some t 1 with t 1 N. Further, for any t R(θ 1 ), one has t t 1 2N and ϕ t θ 1 = ϕ t t 1 θ B. (3.5)

20 20 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Since θ B, Claim 2 shows that (3.5) can only hold provided t t 1 < s, that is, t (t 1 s, t 1 + s). Since t was chosen arbitrarily in R(θ 1 ), this shows R(θ 1 ) (t 1 s, t 1 +s). Thus, m(r(θ 1 )) 2s. This completes the proof of the lemma. We now proceed with the proof of the Spectral Mapping Theorem. Theorem 3.3. Let Γ be the generator of the evolutionary semigroup (2.2). Then e tσ(γ) = σ(t t ) \ {0}, t > 0. (3.1) Moreover, the spectrum σ(γ) is invariant with respect to translations along the imaginary axis, and the spectrum σ(t t ), t > 0, is invariant with respect to rotations centered at origin. Proof. As noted in the beginning of this section, to prove (3.1) it suffices to show: 1 σ ap (T ) implies 0 σ ap (Γ). In fact, we show that σ ap (Γ) contains the entire imaginary axis whenever 1 σ ap (T ). Since σ ap (Γ) contains the boundary of σ(γ), all assertions of the theorem follow from this. Let 1 σ ap (T ), and let ξ R. We begin by using Lemma 3.1 with A = T to obtain, for any N 2, a function f C 0 (Θ, X) satisfying: (3.6a) f C0 (Θ,X) = 1; (3.6b) T N f f C0 (Θ,X) 1 8 ; (3.6c) T k f C0 (Θ,X) 2, for k = 0, 1,..., 2N. For this f, fix s (0, 1) such that and at the same time T t+n f T N f C0 (Θ,X) 1, for t s, (3.7) 16 e iξt 1 1, for t s. (3.8) 32 With the goal of constructing an approximate eigenfunction g, for Γ, corresponding to iξ, choose a smooth function γ : R [0, 1] such that: (3.9a) γ(t) = 0, for t / (0, 2N); (3.9b) γ (t) 2, for all t R; N (3.9c) γ(t) = 1, for t N s. Now fix θ 0 Θ, a nonperiodic point of ϕ t which satisfies f(θ 0 ) 7 8 f C 0 (Θ,X) = 7 8. (3.10)

21 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 21 Set θ N := ϕ N θ 0, and use Lemma 3.2 to obtain an open neighborhood, B, of θ N and a bump function α with the properties a) c) listed there. Define g C 0 (Θ, X) by g(θ) = e iξt γ(t)(t t αf)(θ) dt, θ Θ. (3.11) Note that due to (3.9a), the integration goes from 0 to 2N. Also, note that Γg = iξg e iξt γ (t)(t t αf) dt. Indeed, Γg = d T τ g = d e iξt γ(t)(t t+τ αf) dt dτ τ=0 dτ τ=0 = d e iξ(t τ) γ(t τ)(t t αf) dt dτ τ=0 [ = iξe iξt γ(t)(t t αf) e iξt γ (t)(t t αf) ] dt. We now proceed to show that Γg iξ = O( 1 ) g with the following two claims. N Claim 1. Set C := max T t L(X). Then Γg iξg C0 (Θ,X) 8Cs 0 t 1 N. Proof of Claim 1. First use (3.9a) and (3.9b) to obtain 2N Γg iξg = e iξt γ (t)α(ϕ t )T t f( ) dt Using (3.6c), note that 0 2 N max T t f max 0 t 2N x Θ 2N 0 α(ϕ t θ) dt. (3.12) max T t f max max T τ T k f 2C. (3.13) 0 t 2N 0 k 2N 0 τ 1 Also, the change of variable t t + N gives max x Θ 2N 0 α(ϕ t θ) dt = max x Θ N N α(ϕ t (ϕ N θ)) dt = max x Θ N N α(ϕ t θ) dt. Now, for fixed θ Θ, recall from Lemma 3.2 that the set R(θ) = {t R : t N, ϕ t θ B} has measure m(r(θ)) 2s. Consider t N; then α(ϕ t θ) 1 for t R(θ), and α(ϕ t θ) = 0 for t / R(θ), and so max θ Θ N N α(ϕ t θ) dt max θ Θ R(θ) dt 2s. (3.14)

22 22 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Using (3.13) and (3.14) in the inequality (3.12) gives This proves Claim 1. Claim 2. g C0 (Θ,X) Γg iξg 2 N s C 2s = 8Cs N. Proof of Claim 2. Recall θ N = ϕ N θ 0. Using (3.9a) and a change of variable t t + N gives: g(θ 0 ) = = 2N 0 N N e iξt γ(t)α(ϕ t θ 0 )(T t f)(θ 0 ) dt e iξ(t+n) γ(t + N)α(ϕ t θ N )(T t+n f)(θ 0 ) dt. And so, Lemma 3.2 b) and (3.9c) show that g(θ 0 ) = s s = e iξn s e iξ(t+n) γ(t + N)α(ϕ t θ N )(T t+n f)(θ 0 ) dt s = e iξn (I 1 + I 2 + I 3 ), where we have denoted for brevity: I 1 = f(θ 0 ) s s I 2 = (T N I)f(θ 0 ) I 3 = s s e iξt α(ϕ t θ N )(T t+n f)(θ 0 ) dt e iξt α(ϕ t θ N ) dt, s s e iξt α(ϕ t θ N ) dt, e iξt α(ϕ t θ N )[(T t+n T N )f](θ 0 ) dt. It follows from Lemma 3.2 a) (and recall 0 α(θ) 1), and (3.8) and (3.10) that s I 1 = f(θ 0 ) e iξt α(ϕ t θ N ) dt s ( s f(θ 0 ) α(ϕ t s ) θ N ) dt (e iξt 1)α(ϕ t θ N ) dt s s ( s 4 f(θ 0 ) α(ϕ t s θ s N ) dt (e iξt 1) ) dt s 4 ( s f(θ 0 ) 2 1 ) 32 2s 7 ( ) 7s = 49s

23 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 23 On the other hand, (3.6b) and (3.7) show that I 2 (T N s I)f e iξt α(ϕ t θ N ) dt 1 8 2s = s 4, I 3 As a result, s s s e iξt α(ϕ t θ 0 ) (T t+n T N )f dt s = s 8. g g(θ 0 ) = I 1 +I 2 +I 3 I 1 I 2 I 3 49s 128 s 4 s 8 = s 128. This proves Claim 2. Returning to the proof of the theorem, we combine Claim 1 with Claim 2 to obtain Γg iξg 8Cs N 8C N 128 g. Since N 2 was arbitrary, this shows iξ σ ap (Γ). 4. Exponential Dichotomy Let {T t } t 0 be an evolutionary semigroup (2.2). In this section, we relate the hyperbolicity of the weighted translation operator T = T 1 = av, where a(θ) = Φ(ϕ 1 θ, 1), to the exponential dichotomy of the LSPF ˆϕ t. One can assume in this section, that t Z. We also relate the hyperbolicity of T to the hyperbolicity of weighted shift operators π θ (T ), θ Θ. The representation π θ of B was introduced in Section 2 (see (2.16)). For each θ Θ, the weighted shift operator π θ (T ) acts on l (Z, X) by the rule π θ (T ) = diag{φ(ϕ n 1 θ, 1)} n Z S, where S : (x n ) n Z (x n 1 ) n Z. It is useful to make the observation that for k Z, π θ (T k ) = [π θ (T )] k = diag{φ(ϕ n k θ, k)} n Z S k = S k diag{φ(ϕ n θ, k)} n Z. (4.1) For x = (x n ) n Z, and ȳ = (y n ) n Z in l (Z, X), note that the equation [I π θ (T )] x = ȳ can be expressed componentwise as x n+1 Φ(ϕ n θ, 1)x n = y n+1, n Z. As seen in Section 5.4 (see also [25, Prop. 3.22]), the hyperbolicity of π θ (T ) is equivalent to the existence of exponential dichotomy for ˆϕ t along the orbit through θ, i.e., pointwise dichotomy (see [16, Theorem 7.6.5] and [8]).

24 24 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH We begin with the definition of exponential dichotomy on Θ (or global dichotomy [8, 9, 16, 15, 28, 48]). Definition 4.1. A linear skew product flow ˆϕ t has exponential dichotomy on Θ if there exists a continuous projection P : Θ L s (X) such that for θ Θ and t 0, a) P (ϕ t θ)φ(θ, t) = Φ(θ, t)p (θ); b) Φ Q (θ, t) is invertible from Im Q(θ) to Im Q(ϕ t θ); c) there exist constants M, β > 0 such that for t > 0 Φ P (θ, t) Me βt, [Φ Q (θ, t)] 1 Me βt. If a projection P : Θ L(X) and a cocycle Φ satisfy a), above, then Φ P (θ, t) and Φ Q (θ, t) will be used to denote the restrictions Φ(θ, t)p (θ): Im P (θ) Im P (ϕ t θ) and Φ(θ, t)q(θ): Im Q(θ) Im Q(ϕ t θ), respectively. The main result of this section follows. As usual, the set of nonperiodic points of ϕ t is assumed to be dense in Θ. Theorem 4.1. The following are equivalent: i) σ(t ) T = in C 0 (Θ, X); ii) σ(π θ (T )) T =, on l (Z, X), for all θ Θ and there exists a constant C > 0, such that [π θ (T ) λi] 1 B(l (Z,X)) C for all θ Θ, λ T; iii) The LSPF ˆϕ t has exponential dichotomy. (4.2) We note (see Remark 3 in Section 5.6) that condition (4.2) often follows automatically from σ(π θ (T )) T =, θ Θ. For results related to i) iii), see [42, 43, 44, 45] where a quite different method was used. We begin the proof of the theorem with an observation about the operators π θ (T ). Proposition 4.2. σ(π θ (T )) is invariant with respect to rotations centered at the origin. Moreover, for σ(π θ (T )) T =, [λ π θ (T )] 1 B(l (Z,X)) = [I π θ (T )] 1 B(l (Z,X)), λ T. Proof. For ω T, define Λ = diag{ω n } n Z. Then Λ is an invertible operator on l (Z, X), and Λπ θ (λ T )Λ 1 = λ ω diag{a(ϕ n θ)} n Z S = ω(ω 1 λ π θ (T )).

25 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 25 Hence, [λ π θ (T )] 1 = [ω 1 λ π θ (T )] 1 and the proposition follows. As a result, statement ii) in Theorem 4.1 can be replaced by the statement ii ) I π θ (T ) is invertible for all θ Θ, and there exists C > 0 such that [I π θ (T )] 1 B(l (Z,X)) C for all θ Θ. The proof of Theorem 4.1 will follow from a series of lemmas. The Proof of i) ii ) is given by the following lemma. Lemma 4.3. If b = I T is an invertible operator on C 0 (Θ, X), then π θ (b) is invertible on l (Z, X) for all θ Θ. Moreover, for all θ Θ. [π θ (b)] 1 B(l (Z,X)) b 1 B(C0 (Θ,X)) Proof. We first show that π θ (b) is injective, uniformly for θ Θ, by showing that for any x l (Z, X), π θ (b) x l (Z,X) ( b 1 B(C0 (Θ,X))) 1 x l (Z,X). (4.3) It suffices to prove (4.3) for finitely supported x = (x n ) N n= N. Let θ 0 Θ be a nonperiodic point of ϕ t. Let ɛ > 0. Since θ a(θ)x is continuous for all x X, there exists δ > 0 such that for B := B(θ 0, δ), θ B implies [a(θ 0 ) a(θ)]x n X < ɛ, for all n N. Moreover, δ can be chosen so that, in addition, ϕ n (B) ϕ k (B) = for all k n, n, k N. (4.4) Choose a continuous function α: Θ [0, 1] such that α(θ 0 ) = 1, and α(θ) = 0 for θ / B. Define f C 0 (Θ, X) by α(ϕ n θ)x n, if θ ϕ n (B), n N f(θ) = 0, otherwise. Since b 1 B(C 0 (Θ, X)), we have b 1 bf f C0 (Θ,X) = sup f(θ) X = x l (Z,X). θ Θ (4.5)

26 26 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Also, since f(ϕ 1 θ) = α(ϕ n θ)x n 1 for θ ϕ n (B), n N, and f(ϕ 1 θ) = 0 otherwise, we have bf C0 (Θ,X) = sup f(θ) a(θ)f(ϕ 1 θ) X θ Θ = sup sup α(ϕ n θ)x n a(θ)α(ϕ n θ)x n 1 X n N θ Θ sup sup x n a(ϕ n θ)x n 1 X n N θ Θ sup sup{ x n a(ϕ n θ 0 )x n 1 X + a(ϕ n θ 0 )x n 1 a(ϕ n θ)x n 1 X } n N θ Θ < π θ0 (b) x l (Z,X) + ɛ, using (4.4) in the last inequality. So, Combining (4.5) and (4.6) shows bf C0 (Θ,X) π θ0 (b) x l (Z,X). (4.6) π θ0 (b) x l (Z,X) bf C0 (Θ,X) 1 b 1 B(C0 (Θ,X)) x l (Z,X). We have shown that (4.3) holds for all nonperiodic points of ϕ t. If θ 0 is a periodic point of ϕ t, then choose a sequence of nonperiodic points {θ k } k=1 in Θ converging to θ 0. Since the map θ π θ (b) x is continuous, (4.3) holds, and hence π θ (b) is uniformly injective for all θ Θ. We now check that π θ (b) is surjective for all θ Θ. Let x l (Z, X). As before, consider x = (x n ) N n= N. First suppose θ 0 is a nonperiodic point of ϕ t. Define f C 0 (Θ, X) as above and note that since b is surjective, there exists a g C 0 (Θ, X) such that bg = f; that is, g(θ) a(θ)g(ϕ 1 θ) = f(θ) for all θ Θ. Substituting θ = ϕ n θ 0 into this equation and defining ȳ l (Z, X) by y n = g(ϕ n θ 0 ) for n N, gives π θ0 (b)ȳ = x. To address the case for which θ 0 is a periodic point of ϕ t, so that ϕ k θ 0 = θ 0 for some k Z, we begin with Case k = 1. Assume ϕθ 0 = θ 0. If T = av is hyperbolic on C 0 (Θ, X), then π θ0 (T ) = diag{a(θ 0 )} n Z S is hyperbolic on l (Z, X). Proof. By hypothesis, I av is surjective on C 0 (Θ, X), and so I a(θ 0 ) is surjective on X. Indeed, given x X, choose f C 0 (Θ, X) with f(θ 0 ) = x and then choose g C 0 (Θ, X) so that (I av )g = f; setting y = g(θ 0 ) gives (I a(θ 0 ))y = x. Therefore, a(θ 0 ) is hyperbolic

27 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 27 on X. As in Lemma 2.4 of [22], this implies that diag{a(θ 0 )} n Z S is hyperbolic on l (Z, X). Case k > 1. Assume ϕ k θ 0 = θ 0. If T is hyperbolic on C 0 (Θ, X), then π θ0 (T ) is hyperbolic on l (Z, X). Proof. By standard spectral properties, it suffices to consider hyperbolicity for T k. First note (see (4.1)) that π θ0 (T k ) = S k diag{φ(ϕ n θ 0, k)} n Z. This can be expressed as the operator Ŝ diag{d} n Z, on l (Z, X k ) l (Z, X), (4.7) where Ŝ, on l (Z, X k ), and d, on X k, are defined as x n+1 x n+k+1 Φ(θ 0, k). and d =... x n+k Φ(ϕ k 1 θ 0, k) n Z. x n+2k n Z As in Lemma 2.4 of [22], (4.7) is hyperbolic on l (Z, X k ) provided d is hyperbolic on X k. Therefore, it suffices to show that Φ(θ 0, k),..., Φ(ϕ k 1 θ 0, k) are hyperbolic on X. Applying Case k = 1 with φ := ϕ k and the representation π θ,φ : a diag{a(φ n θ)} n Z shows that π θ0,φ(t k ) = S k diag{φ(ϕ kn θ 0, k)} n Z = S k diag{φ(θ 0, k)} n Z is hyperbolic. As before, expressing this operator as Ŝ diag{φ(θ 0, k)} n Z on l (Z, X k ) and applying Lemma 2.4 of [22] shows that Φ(θ 0, k) is hyperbolic. This proves the case k > 1. Thus, the lemma is proved. The Proof of ii) i) requires showing that the invertibility of b in B can be derived from the invertibility of all its images π θ (b) in B(l (Z, X)). To do this, we consider the algebra D of operators in B(l (Z, X)) as defined in Section 2. The next lemma proves a slightly stronger statement than ii) i). Lemma 4.4. Let b = λi T. Assume π θ (b) is invertible in B(l (Z, X)) and that there exists a C > 0 such that [π θ (b)] 1 B(l (Z,X)) C for all θ Θ. Then b 1 exists and is an element of B.

28 28 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Proof. Let b = I T ; the proof remains the same for b = λi T, λ T. Using the notation of Proposition 2.12, set d(θ) := π θ (b) = I π θ (a)s. Since sup θ Θ a(θ) L(X) <, it follows that k sup θ Θ d k (θ) B(l (Z,X)) = 1 + sup π θ (a) B(l (Z,X)) <, θ Θ Conse- and hence d(θ) satisfies the conditions of Proposition quently, for each θ, π θ (b) = d(θ) has an inverse that is in D: [π θ (b)] 1 = c k (θ)s k, for some c k (θ) = diag{c (n) k (θ)} n Z in C, k Z. Moreover, since sup θ Θ π θ (b) B(l (Z,X)) C, Proposition 2.12, shows that sup c k (θ) B(l (Z,X)) <. θ Θ k The lemma is proved by showing that for each k Z, θ c (0) k (θ) is a continuous, bounded function from Θ L s(x), (4.8) (in the notation of Section 2, c (0) k A) and then showing that the operator r := c (0) k V k is in B and satisfies r = b 1. Let k Z, and fix x X and θ 0 Θ. Define x = (x n ) n Z in l (Z, X) by x n = x if n = k, and x n = 0 if n k. Then for any θ Θ, [c (0) k (θ) c(0) k (θ 0)]x X = [c (0) j (θ) c (0) j (θ 0 )]x j j= X sup [c (n) j (θ) c (n) j (θ 0 )]x n j n Z j= X = diag{c (n) j j= = [c j (θ) c j (θ 0 )]S j x j= = [π θ (b)] 1 [π θ0 (b) π θ (b)][π θ0 (b)] 1 x C [π θ0 (b) π θ (b)]ȳ l (Z,X), (θ) c (n) j (θ 0 )} n Z S j x l (Z,X) l (Z,X) l (Z,X)

29 EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LSPF 29 where the last inequality comes from letting ȳ = [π θ0 (b)] 1 x and using the assumption [π θ (b)] 1 C. Since the map θ π θ (b)ȳ is continuous for any ȳ l (Z, X), c (0) k is continuous at θ 0. Moreover, note that c k (θ) = diag{c (n) k (θ)} n Z satisfies, for each n Z, the inequality c k (θ) B(l (Z,X)) c (n) k (θ) L(X). Therefore, k sup θ Θ c k (θ) B(l (Z,X)) < implies, in particular, that sup θ Θ c (0) k (θ) L(X) <, and so each c (0) k is bounded. Thus, (4.8) holds. Also, since c (0) k B(C 0 (Θ,X)) = sup θ Θ c (0) k (θ) L(X), it follows that c (0) k B(C 0 (Θ,X)) = sup θ Θ c (0) k (θ) L(X) Therefore, the operator r := c (0) k V k is in B. Now observe that c (n) k sup c k (θ) B(l (Z,X)) <. θ Θ (θ) = c(0) k (ϕn θ), n Z. (4.9) Indeed, for any n Z, π θ (a)s n = S n π ϕ n θ(a), and so Therefore, π ϕ n θ(b) = I diag{a(ϕ n+k θ)} k Z S = S n π θ (b)s n. [π ϕ n θ(b)] 1 = S n [π θ (b)] 1 S n = S n = On the other hand, [π ϕ n θ(b)] 1 = diag{c (i+n) k (θ)} i Z S k. c k (ϕ n θ)s k = diag{c (i) k (θ)} i ZS k S n diag{c (i) k (ϕn θ)} i Z S k, so (4.9) holds. As a consequence, rb = br = I. Indeed, using (4.9), π θ (r) = = π θ (c (0) k )Sk = diag{c (n) k (θ)} n ZS k = diag{c (0) k (ϕn θ)} n Z S k c k (θ)s k = [π θ (b)] 1.