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-9400518 and by the Summer Research Fellowship of the University of Missouri.

Supported by the National Science Foundation under the grant DMS-9201357. Supported by the University of Missouri Research Board.

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

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 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 φ̂^{t} is defined by the following formula (see Section 2)

φ̂^{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

\frac{dx}{dt} = A(φ^{t}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

\frac{dx}{dt} = A(t)x(t), t ∈ ℝ;  (1.2)

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)

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 φ̂^{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) = − \frac{d}{dt}f ∘ φ^{t}(θ)|_{t=0}+A(θ)f(θ), and (Γf)(τ) = −\frac{df}{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 φ̂^{t}-invariant splitting Θ × X = X_{P} + X_{Q} with uniform exponential decay of Φ_{P} on X_{P} and of Φ_{Q}^{−1} 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_{Q}^{−1} 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 φ̂^{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 φ̂^{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 φ̂^{t} along the trajectory through θ. These operators were, in fact, introduced in [16] for the line setting, and in

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 φ̂^{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 φ̂^{t} (see [47]) is the set of all λ ∈ ℝ such that the LSPF, φ̂_{λ}^{t}, 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 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

φ̂^{t}: Θ × X → Θ × X,  φ̂^{t}(θ, x) = (φ^{t}θ, Φ(θ, t)x),  t ≥ 0.(2.1)

We note that for exponentially bounded cocycles, the map φ̂^{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.

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 ϵ > 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∥ ≤ ϵ/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_{θ∈Θ} ∥T^{t}f(θ) − T^{t_{0}}f(θ)∥ < ϵ/3.

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})∥ < ϵ/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

\begin{aligned}∥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})∥ < ϵ/3. \end{aligned}

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})∥ < ϵ/3.

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∥ < ϵ. □

2.2. 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

\frac{dx}{dt} = A(φ^{t}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) = \frac{d}{dt}Φ(θ, t)|_{t=0}. Denote (df)(θ) =

\frac{d}{dt}f ∘ φ^{t}(θ)|_{t=0}. The generator Γ of the group (2.2) is given as follows:

(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].

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′_{0}f = −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 + ∫_{0}^{t} 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 + ∫_{0}^{t} e^{(t−τ)Γ_{0}}A_{1}S^{τ}f dτ, f ∈ C_{0}(Θ, X), t ≥ 0. (2.9)

We show that S^{t} = T^{t}. Indeed, for f ∈ C_{0}(Θ, X) define g = ∫_{0}^{t} e^{(t−τ)Γ_{0}}A_{1}T^{τ}f dτ. Then (2.8) implies

g(θ) = ∫_{0}^{t} 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).

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

\frac{dx}{dt} = A(t)x(t), t ∈ ℝ.  (2.10)

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) = −\frac{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′ = −\frac{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.

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]):

\frac{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) = −\frac{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. (2.13)

Then Γf = Γ′f. Indeed

(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,

\frac{d}{dt}(T^{t}f)(τ)|_{t=0} = −α′(τ)U(τ, s)x_{s}, τ ∈ ℝ.

On the other hand, since τ ↦ U(τ, s)x_{s} satisfies (2.11), one has:

\frac{df}{dτ} = α′(τ)U(τ, s)x_{s}+α(τ)\frac{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 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 ϵ > 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∥ ≤ ϵ/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}}_{j=1}^{n}. For I_{j} := (s_{j}, s′_{j}), one has ∥U(τ, s_{j})v − v∥ ≤ ϵ/2 for τ ∈ I_{j}, j = 1, …, n.

Consider a smooth partition of unity {γ_{j}}_{j=1}^{n} for {I_{j}}_{j=1}^{n}, that is, smooth functions γ_{j} : ℝ → [0, 1] such that

∑_{j=1}^{n} γ_{j}(τ) = 1 for τ ∈ supp β, and supp γ_{j} ⊂ I_{j}, j = 1, …, n.

Since the D_{s_{j}} are dense in X, one can choose v_{j} ∈ D_{s_{j}} such that ∥v − v_{j}∥ ≤ ϵ/2.

Define the function

h(τ) := ∑_{j=1}^{n} β(τ)γ_{j}(τ)U(τ, s_{j})v_{j}, τ ∈ ℝ.

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)∥ = ∥β(τ) ∑_{j=1}^{n} γ_{j}(τ)v − ∑_{j=1}^{n} β(τ)γ_{j}(τ)U(τ, s_{j})v_{j}∥

≤ |β(τ)| (∑_{j=1}^{n} γ_{j}(τ)∥v − v_{j}∥ + ∑_{j=1}^{n} γ_{j}(τ)∥v_{j} − U(τ, s_{j})v_{j}∥)
≤ ϵ max_{τ} |β(τ)|,

and the proof is completed. □

2.3. 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)}.

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(C_{0}(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 = ∑_{k=−∞}^{∞} a_{k}V^{k}, where a_{k} ∈ 𝔄, and ∥b∥_{1} := ∑_{k=−∞}^{∞} ∥a_{k}∥ < ∞.

(2.14)

Proposition 2.10. 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} = ∑_{k=−N}^{N} a_{k}V^{k} in B(C_{0}(Θ, X)), where a_{k} ∈ 𝔄 and N ∈ ℕ,

∥b_{N}∥_{B(C_{0}(Θ,X))} ≥ ∥a_{k}∥_{B(C_{0}(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 ϵ > 0. For a ∈ 𝔄, ∥a∥_{B(C_{0}(Θ,X))} = sup_{θ∈Θ} sup_{∥x∥_{X}=1} ∥a(θ)x∥_{X},

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(C_{0}(Θ,X))} − ϵ.

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

f(θ) = \begin{cases}α(φ^{−k}θ)x, if θ ∈ φ^{k}(B), |k| ≤ N \\ 0 otherwise. \end{cases}

Then ∥f∥_{C_{0}(Θ,X)} = 1 and

∥b∥_{B(C_{0}(Θ,X))} ≥ ∥bf∥_{C_{0}(Θ,X)} = max_{θ∈Θ} ∥ ∑_{k=−N}^{N} a_{k}(θ)α(φ^{−k}θ)x∥_{X}

= max_{θ∈Θ} max_{|k|≤N} ∥a_{k}(θ)α(φ^{−k}θ)x∥_{X} ≥ ∥a_{0}(θ_{0})x∥_{X}

≥ ∥a_{0}∥_{B(C_{0}(Θ,X))} − ϵ.

□

16 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Proposition 2.11. (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)} = ∑_{k=−∞}^{∞} a_{k}^{(n)}V^{k}.

Set a_{k} = ∥ · ∥_{1}- lim_{m→∞} a_{k}^{(m)}, k ∈ ℤ. For every m, n in ℤ,

| ∑_{k=−∞}^{∞} ∥a_{k}^{(m)}∥ − ∑_{k=−∞}^{∞} ∥a_{k}^{(n)}∥| ≤ ∥b^{(m)} − b^{(n)}∥_{1}.

So a sequence (∥a_{k}^{(m)}∥)_{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 ϵ > 0 and for sufficiently large m, n, the inequality

ϵ ≥ ∥b^{(m)} − b^{(n)}∥_{1} = ∑_{k=−∞}^{∞} ∥a_{k}^{(m)} − a_{k}^{(n)}∥ ≥ ∑_{k=−N}^{N} ∥a_{k}^{(m)} − a_{k}^{(n)}∥

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) = ∑_{k=−∞}^{∞} 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 = ∑_{k=−∞}^{∞} d_{k}S^{k} where d_{k} ∈ ℭ, and ∥d∥_{1} := ∑_{k=−∞}^{∞} ∥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].

Proposition 2.12. Let d(θ) = ∑_{k=−∞}^{∞} d_{k}(θ)S^{k} be an operator in 𝔇, and

assume that

∑_{k=−∞}^{∞} sup_{θ∈Θ} ∥d_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} < ∞.

If d(θ)^{−1} ∈ B(ℓ_{∞}(ℤ, X))) for each θ ∈ Θ and if for some C > 0,

sup_{θ∈Θ} ∥d(θ)^{−1}∥_{B(ℓ_{∞}(ℤ,X))} < C,

then d(θ)^{−1} ∈ 𝔇 for each θ ∈ Θ. Moreover, if d(θ)^{−1} = ∑_{k=−∞}^{∞} c_{k}(θ)S^{k},

then

∑_{k=−∞}^{∞} sup_{θ∈Θ} ∥c_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} < ∞.

3. Spectral Mapping Theorem

In this section we prove that the spectral mapping property

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 ϵ-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 ⊆ ∪_{j=0}^{2N}φ^{j}(D) and ∥T f − f∥ = O(\frac{1}{N}). Using f, we construct a function g ∈ C_{0}(Θ, X) such that ∥Γg∥ = O(\frac{1}{N})∥g∥, hence 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 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} ≤ \frac{1}{8};

b) ∥A^{k}x∥_{X} ≤ 2 for k = 0, 1, …, 2N.

Proof. Set c = ∑_{k=0}^{2N} ∥A∥^{k}. From the identity A^{k} − I = (A^{k−1} + … + 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∥ ≤ \frac{1}{8c}. Set k = N in (3.2) to obtain a). Moreover, (3.2) shows that for k = 0, 1, …, 2N,

∥A^{k}x∥ ≤ ∥(A^{k} − I)x∥ + ∥x∥ ≤ c∥(A − I)x∥ + 1 ≤ c\frac{1}{8c} + 1 ≤ 2.

□

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| ≤ \frac{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 ϵ_{∗} > 0, and all ϵ ≤ ϵ_{∗}, if δ < ϵ, then θ ∈ B′ := B(θ_{0}, δ) implies φ^{t}θ ∉ B′ provided \frac{s}{2} ≤ |t| ≤ 5N.

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 \frac{s}{2} ≤ |t_{n}| ≤ 5N, one has φ^{t_{n}}θ_{n} ∈ B_{n}. Passing to a subsequence, one can assume t_{n} → t_{∗} for some t_{∗} with \frac{s}{2} ≤ |t_{∗}| ≤ 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

EVOLUTIONARY SEMIGROUPS AND  DICHOTOMY OF LSPF  19

φ^{t_{n}}θ_{n} → φ^{t_{∗}}θ_{0}. On the other hand, φ^{t_{n}}θ_{n} ∈ B_{n} implies φ^{t_{n}}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 < ϵ_{∗}, and set B′ = B(θ_{0}, δ) as in Claim 1. Choose an open neighborhood B′′ of θ_{0} such that \overline{B′′} ⊂ B′. Denote:

B := ⋃_{|t|≤ \frac{s}{4}} φ^{t}(B′), C := ⋃_{|t|≤ \frac{s}{4}} φ^{t}(B′′).  (3.3)

Since \overline{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| ≤ \frac{s}{4}\} ⊂ C and so α(φ^{t}θ_{0}) = 1 whenever |t| ≤ \frac{s}{4}. This proves a).

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}| ≤ \frac{s}{4}. This implies

φ^{t_{∗}+t_{1}−t_{2}}θ_{1} ∈ B′.  (3.4)

Note that

|t_{∗} + t_{1} − t_{2}| ≤ 2N + \frac{s}{4} + \frac{s}{4} ≤ 5N

and

|t_{∗} + t_{1} − t_{2}| ≥ |t_{∗}| − |t_{1}| − |t_{2}| ≥ s − \frac{s}{4} − \frac{s}{4} = \frac{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 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∥_{C_{0}(Θ,X)} = 1;

(3.6b) ∥T^{N}f − f∥_{C_{0}(Θ,X)} ≤ \frac{1}{8};
(3.6c) ∥T^{k}f∥_{C_{0}(Θ,X)} ≤ 2, for k = 0, 1, …, 2N.

For this f, fix s ∈ (0, 1) such that

∥T^{t+N}f − T^{N}f∥_{C_{0}(Θ,X)} ≤ \frac{1}{16}, for |t| ≤ s,  (3.7)

and at the same time

|e^{−iξt} − 1| ≤ \frac{1}{32}, for |t| ≤ s.  (3.8)

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)| ≤ \frac{2}{N} , for all t ∈ ℝ;
(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})∥ ≥ \frac{7}{8}∥f∥_{C_{0}(Θ,X)} = \frac{7}{8}.  (3.10)

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 = \frac{d}{dτ} |_{τ=0} T^{τ}g = \frac{d}{dτ} |_{τ=0} ∫_{−∞}^{∞} e^{−iξt}γ(t)(T^{t+τ}αf) dt

= \frac{d}{dτ} |_{τ=0} ∫_{−∞}^{∞} e^{−iξ(t−τ)}γ(t − τ)(T^{t}αf) dt

= ∫_{−∞}^{∞} [iξe^{−iξt}γ(t)(T^{t}αf) − e^{−iξt}γ′(t)(T^{t}αf)] dt.

We now proceed to show that ∥Γg − iξ∥ = O(\frac{1}{N})∥g∥ with the following two claims.

Claim 1 . Set C := max_{0≤t≤1} ∥T^{t}∥_{L(X)}. Then ∥Γg − iξg∥_{C_{0}(Θ,X)} ≤ \frac{8Cs}{N}.

Proof of Claim 1. First use (3.9a) and (3.9b) to obtain

∥Γg − iξg∥ = ∥− ∫_{0}^{2N} e^{−iξt}γ′(t)α(φ^{−t}·)T^{t}f(·) dt∥

≤ \frac{2}{N} · max_{0≤t≤2N} ∥T^{t}f∥ · max_{x∈Θ} ∫_{0}^{2N} α(φ^{−t}θ) dt. (3.12)

Using (3.6c), note that

max_{0≤t≤2N} ∥T^{t}f∥ ≤ max_{0≤k≤2N} max_{0≤τ≤1} ∥T^{τ}T^{k}f∥ ≤ 2C.  (3.13)

Also, the change of variable t ↦ −t + N gives

max_{x∈Θ} ∫_{0}^{2N} 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)

Using (3.13) and (3.14) in the inequality (3.12) gives

∥Γg − iξg∥ ≤ \frac{2}{N} · 2C · 2s = \frac{8Cs}{N}.

This proves Claim 1. □

Claim 2 . ∥g∥_{C_{0}(Θ,X)} ≥ \frac{s}{128}.

Proof of Claim 2. Recall θ_{N} = φ^{−N}θ_{0}. Using (3.9a) and a change of variable t ↦ t + N gives:

\begin{aligned}g(θ_{0}) & = ∫_{0}^{2N} e^{−iξt}γ(t)α(φ^{−t}θ_{0})(T^{t}f)(θ_{0}) dt \\ & = ∫_{−N}^{N} e^{−iξ(t+N)}γ(t + N)α(φ^{−t}θ_{N})(T^{t+N}f)(θ_{0}) dt. \end{aligned}

And so, Lemma 3.2 b) and (3.9c) show that

g(θ_{0}) = ∫_{−s}^{s} e^{−iξ(t+N)}γ(t + N)α(φ^{−t}θ_{N})(T^{t+N}f)(θ_{0}) dt
= e^{−iξN} ∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N})(T^{t+N}f)(θ_{0}) dt
= e^{−iξN}(I_{1} + I_{2} + I_{3}),

where we have denoted for brevity:

\begin{aligned}I_{1} & = f(θ_{0}) ∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N}) dt, \\ I_{2} & = (T^{N} − I)f(θ_{0}) ∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N}) dt, \\ I_{3} & = ∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N})[(T^{t+N} − T^{N})f](θ_{0}) dt. \end{aligned}

It follows from Lemma 3.2 a) (and recall 0 ≤ α(θ) ≤ 1), and (3.8) and (3.10) that

∥I_{1}∥ = ∥f(θ_{0})∥ |∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N}) dt|

≥ ∥f(θ_{0})∥ (|∫_{−s}^{s} α(φ^{−t}θ_{N}) dt| − |∫_{−s}^{s}(e^{−iξt} − 1)α(φ^{−t}θ_{N}) dt|)

≥ ∥f(θ_{0})∥ (|∫ − \frac{s}{4}_{\frac{s}{4}} α(φ^{−t}θ_{N}) dt| − ∫_{−s}^{s} |(e^{−iξt} − 1)| dt)

≥ ∥f(θ_{0})∥ (\frac{s}{2} − \frac{1}{32}2s) ≥ \frac{7}{8} (\frac{7s}{16}) = \frac{49s}{128}.

On the other hand, (3.6b) and (3.7) show that

\begin{aligned}∥I_{2}∥ & ≤ ∥(T^{N} − I)f∥ |∫_{−s}^{s} e^{−iξt}α(φ^{−t}θ_{N}) dt| ≤ \frac{1}{8}2s = \frac{s}{4}, \\ ∥I_{3}∥ & ≤ ∫_{−s}^{s} |e^{−iξt}α(φ^{−t}θ_{0})| ∥(T^{t+N} − T^{N})f∥ dt ≤ \frac{1}{16}2s = \frac{s}{8}. \end{aligned}

As a result,

∥g∥ ≥ ∥g(θ_{0})∥ = ∥I_{1}+I_{2}+I_{3}∥ ≥ ∥I_{1}∥−∥I_{2}∥−∥I_{3}∥ ≥ \frac{49s}{128}−\frac{s}{4}−\frac{s}{8} = \frac{s}{128}.

This proves Claim 2. □

Returning to the proof of the theorem, we combine Claim 1 with Claim 2 to obtain

∥Γg − iξg∥ ≤ \frac{8Cs}{N} ≤ \frac{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 φ̂^{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 x̄ = (x_{n})_{n∈ℤ}, and ȳ = (y_{n})_{n∈ℤ} in ℓ_{∞}(ℤ, X), note that the equation [I − π_{θ}(T)]x̄ = 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 φ̂^{t} along the orbit through θ, i.e., “pointwise” dichotomy (see [16, Theorem 7.6.5] and [8]).

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 φ̂^{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; (4.2)

iii) The LSPF φ̂^{t} has exponential dichotomy. 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)).

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,

∥[π_{θ}(b)]^{−1}∥_{B(ℓ_{∞}(ℤ,X))} ≤ ∥b^{−1}∥_{B(C_{0}(Θ,X))}

for all θ ∈ Θ.

Proof. We first show that π_{θ}(b) is injective, uniformly for θ ∈ Θ, by showing that for any x̄ ∈ ℓ_{∞}(ℤ, X),

∥π_{θ}(b)x̄∥_{ℓ_{∞}(ℤ,X)} ≥ (∥b^{−1}∥_{B(C_{0}(Θ,X))})^{−1}∥x̄∥_{ℓ_{∞}(ℤ,X)}.  (4.3)

It suffices to prove (4.3) for finitely supported x̄ = (x_{n})_{n=−N}^{N}.

Let θ_{0} ∈ Θ be a nonperiodic point of φ^{t}. Let ϵ > 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} < ϵ, for all |n| ≤ N. (4.4)

Moreover, δ can be chosen so that, in addition,

φ^{n}(B) ∩ φ^{k}(B) = ∅ for all k ≠ n, |n|, |k| ≤ 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

f(θ) = \begin{cases}α(φ^{−n}θ)x_{n}, if θ ∈ φ^{n}(B), |n| ≤ N \\ 0, otherwise. \end{cases}

Since b^{−1} ∈ B(C_{0}(Θ, X)), we have

∥b^{−1}∥ · ∥bf∥ ≥ ∥f∥_{C_{0}(Θ,X)} = sup_{θ∈Θ} ∥f(θ)∥_{X} = ∥x̄∥_{ℓ_{∞}(ℤ,X)}. (4.5)

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∥_{C_{0}(Θ,X)} = sup_{θ∈Θ} ∥f(θ) − a(θ)f(φ^{−1}θ)∥_{X}

= sup_{|n|≤N} sup_{θ∈Θ} ∥α(φ^{−n}θ)x_{n} − a(θ)α(φ^{−n}θ)x_{n−1}∥_{X}

≤ sup_{|n|≤N} sup_{θ∈Θ} ∥x_{n} − a(φ^{n}θ)x_{n−1}∥_{X}

≤ sup_{|n|≤N} sup_{θ∈Θ}{∥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}}

< ∥π_{θ_{0}}(b)x̄∥_{ℓ_{∞}(ℤ,X)} + ϵ,

using (4.4) in the last inequality. So,

∥bf∥_{C_{0}(Θ,X)} ≤ ∥π_{θ_{0}}(b)x̄∥_{ℓ_{∞}(ℤ,X)}.  (4.6)

Combining (4.5) and (4.6) shows

∥π_{θ_{0}}(b)x̄∥_{ℓ_{∞}(ℤ,X)} ≥ ∥bf∥_{C_{0}(Θ,X)} ≥ \frac{1}{∥b^{−1}∥_{B(C_{0}(Θ,X))}} ∥x̄∥_{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)x̄ 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 x̄ ∈ ℓ_{∞}(ℤ, X). As before, consider x̄ = (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_{θ_{0}}(b)ȳ = x̄.

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_{θ_{0}}(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

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_{θ_{0}}(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_{θ_{0}}(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} )_{n∈ℤ} ↦ ( x_{n+k+1} ; ⋮ ; x_{n+2k} )_{n∈ℤ} and d = [ \begin{matrix}Φ(θ_{0}, k) \\ & . & . & . \\ & & & & Φ(φ^{k−1}θ_{0}, k) \end{matrix} ] .

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_{θ_{0},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 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_{θ∈Θ} ∥π_{θ}(a)∥_{B(ℓ_{∞}(ℤ,X))} < ∞,

and hence d(θ) satisfies the conditions of Proposition 2.12. Consequently, for each θ, π_{θ}(b) = d(θ) has an inverse that is in 𝔇:

[π_{θ}(b)]^{−1} = ∑_{k=−∞}^{∞} c_{k}(θ)S^{k}, for some c_{k}(θ) = diag{c_{k}^{(n)}(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

∑_{k} sup_{θ∈Θ} ∥c_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} < ∞.

The lemma is proved by showing that for each k ∈ ℤ,

θ ↦ c_{k}^{(0)}(θ) is a continuous, bounded function from Θ → L_{s}(X), (4.8)

(in the notation of Section 2, c_{k}^{(0)} ∈ 𝔄) and then showing that the operator r := ∑_{k=−∞}^{∞} c_{k}^{(0)}V^{k} is in 𝔅 and satisfies r = b^{−1}.

Let k ∈ ℤ, and fix x ∈ X and θ_{0} ∈ Θ. Define x̄ = (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_{k}^{(0)}(θ) − c_{k}^{(0)}(θ_{0})]x∥_{X} = ∥ ∑_{j=−∞}^{∞}[c_{j}^{(0)}(θ) − c_{j}^{(0)}(θ_{0})]x_{−j}∥_{X}

≤ sup_{n∈ℤ} ∥ ∑_{j=−∞}^{∞}[c_{j}^{(n)}(θ) − c_{j}^{(n)}(θ_{0})]x_{n−j}∥_{X}

= ∥( ∑_{j=−∞}^{∞} diag{c_{j}^{(n)}(θ) − c_{j}^{(n)}(θ_{0})}_{n∈ℤ}S^{j}) x̄∥_{ℓ_{∞}(ℤ,X)}

= ∥( ∑_{j=−∞}^{∞}[c_{j}(θ) − c_{j}(θ_{0})]S^{j}) x̄∥_{ℓ_{∞}(ℤ,X)}

= ∥[π_{θ}(b)]^{−1}[π_{θ_{0}}(b) − π_{θ}(b)][π_{θ_{0}}(b)]^{−1}x̄∥_{ℓ_{∞}(ℤ,X)}

≤ C ∥[π_{θ_{0}}(b) − π_{θ}(b)]ȳ∥_{ℓ_{∞}(ℤ,X)} ,

EVOLUTIONARY SEMIGROUPS AND  DICHOTOMY OF LSPF  29

where the last inequality comes from letting ȳ = [π_{θ_{0}}(b)]^{−1}x̄ 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_{k}^{(0)} is continuous at θ_{0}.

Moreover, note that c_{k}(θ) = diag{c_{k}^{(n)}(θ)}_{n∈ℤ} satisfies, for each n ∈ ℤ, the inequality

∥c_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} ≥ ∥c_{k}^{(n)}(θ)∥_{L(X)}.

Therefore, ∑_{k} sup_{θ∈Θ} ∥c_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} < ∞ implies, in particular, that sup_{θ∈Θ} ∥c_{k}^{(0)}(θ)∥_{L(X)} < ∞, and so each c_{k}^{(0)} is bounded. Thus, (4.8) holds.

Also, since ∥c_{k}^{(0)}∥_{B(C_{0}(Θ,X))} = sup_{θ∈Θ} ∥c_{k}^{(0)}(θ)∥_{L(X)}, it follows that

∑_{k=−∞}^{∞} ∥c_{k}^{(0)}∥_{B(C_{0}(Θ,X))} = ∑_{k=−∞}^{∞} sup_{θ∈Θ} ∥c_{k}^{(0)}(θ)∥_{L(X)} ≤ ∑_{k=−∞}^{∞} sup_{θ∈Θ} ∥c_{k}(θ)∥_{B(ℓ_{∞}(ℤ,X))} < ∞.

Therefore, the operator r := ∑_{k=−∞}^{∞} c_{k}^{(0)}V^{k} is in 𝔅.

Now observe that

c_{k}^{(n)}(θ) = c_{k}^{(0)}(φ^{n}θ), n ∈ ℤ.  (4.9)

Indeed, for any n ∈ ℤ, π_{θ}(a)S^{n} = S^{n}π_{φ^{n}θ}(a), and so

π_{φ^{n}θ}(b) = I − diag{a(φ^{n+k}θ)}_{k∈ℤ}S = S^{−n}π_{θ}(b)S^{n}.

Therefore,

[π_{φ^{n}θ}(b)]^{−1} = S^{−n}[π_{θ}(b)]^{−1}S^{n} = S^{−n} ( ∑_{k=−∞}^{∞} diag{c_{k}^{(i)}(θ)}_{i∈ℤ}S^{k}) S^{n}

= ∑_{k=−∞}^{∞} diag{c_{k}^{(i+n)}(θ)}_{i∈ℤ}S^{k}.

On the other hand,

[π_{φ^{n}θ}(b)]^{−1} = ∑_{k=−∞}^{∞} c_{k}(φ^{n}θ)S^{k} = ∑_{k=−∞}^{∞} diag{c_{k}^{(i)}(φ^{n}θ)}_{i∈ℤ}S^{k},

so (4.9) holds.

As a consequence, rb = br = I. Indeed, using (4.9),

π_{θ}(r) = ∑_{k=−∞}^{∞} π_{θ}(c_{k}^{(0)})S^{k} = ∑_{k=−∞}^{∞} diag{c_{k}^{(0)}(φ^{n}θ)}_{n∈ℤ}S^{k}

= ∑_{k=−∞}^{∞} diag{c_{k}^{(n)}(θ)}_{n∈ℤ}S^{k} = ∑_{k=−∞}^{∞} c_{k}(θ)S^{k} = [π_{θ}(b)]^{−1}.

30 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Therefore, I = π_{θ}(r)π_{θ}(b) = π_{θ}(rb), and so π_{θ}(rb − I) = 0 for all θ. One checks directly that for π_{θ}: 𝔅 → 𝔇, ⋂_{θ∈Θ} Ker π_{θ} = {0} and so rb = I. □

The proof i) ⇒ iii) uses the following corollary to Lemma 4.4. Corollary 4.5. If σ(T) ∩ 𝕋 = ∅, then the Riesz projection, P, corresponding to σ(T) ∩ 𝔻 has the form Pf(θ) = P (θ)f(θ), f ∈ C_{0}(Θ, X), for some bounded, continuous projection-valued function P : Θ → L_{s}(X).

Proof. If σ(T) ∩ 𝕋 = ∅, then statement ii) of Theorem 4.1 holds, and so Lemma 4.4 applies and shows (λI − T)^{−1} is in 𝔅 for all λ ∈ 𝕋. Consequently, P = \frac{1}{2πi} ∫_{𝕋}(λI − T)^{−1} dλ is an element of 𝔅. Proceeding exactly as in [25] or [21, Lemma 3], one can show that P ∈ 𝔄. □

Before proceeding, we make an additional observation. Proposition 4.6. Assume σ(T) ∩ 𝕋 = ∅, and let P be the Riesz projection corresponding to σ(T) ∩ 𝔻, where Pf(θ) = P (θ)f(θ), as above. Then the Riesz projection P_{θ} in B(ℓ_{∞}(ℤ, X)) corresponding to σ(π_{θ}(T)) ∩ 𝔻 is given by

π_{θ}(P) = diag{P (φ^{n}θ)}_{n∈ℤ}.

Proof. As in the previous corollary, P ∈ 𝔅. Moreover, π_{θ}((λI−T)^{−1}) = [λI−π_{θ}(T)]^{−1} is in 𝔇, for λ ∈ 𝕋, and the Riesz projection corresponding to σ(π_{θ}(T)) ∩ 𝔻, given by P_{θ} = \frac{1}{2πi} ∫_{𝕋}[λI − π_{θ}(T)]^{−1} dλ, is in 𝔇. Since π_{θ}: 𝔅 → 𝔇 is a continuous homomorphism, P_{θ} = π_{θ}(P). Since P is given by P (·), if follows that P_{θ} = π_{θ}(P) = diag{P (φ^{n}θ)}_{n∈ℤ}. □

The Proof of i) ⇒ iii) is a consequence of the next lemma. Lemma 4.7. If σ(T) ∩ 𝕋 = ∅, then the LSPF φ̂^{t} has exponential dichotomy.

Proof. Let P be the Riesz projection corresponding to σ(T) ∩ 𝔻. By Corollary 4.5, P is given by (Pf)(θ) = P (θ)f(θ) for some projectionvalued function P: Θ → X. We show that P satisfies the properties of Definition 4.1. As already shown, the statement σ(T) ∩ 𝕋 = ∅ implies statement ii) of Theorem 4.1. We use the latter to show that Φ(θ, t) is invertible as an operator from Im Q(θ) to Im Q(φ^{t}θ).

EVOLUTIONARY SEMIGROUPS AND  DICHOTOMY OF LSPF  31

Fix θ ∈ Θ. Statement ii) implies that π_{θ}(T)∩𝕋 = ∅, and Proposition 4.6 shows that the corresponding Riesz projection is given by P_{θ} = π_{θ}(P) = diag{P (φ^{n}θ)}_{n∈ℤ}. Set Q_{θ} = I − P_{θ}, and let k ∈ ℤ. Since π_{θ}(T)Q_{θ} is invertible on Im Q_{θ}, so is [π_{θ}(T)Q_{θ}]^{k} = π_{θ}(T^{k})Q_{θ}. Hence, for any ȳ = (y_{n})_{n∈ℤ} ∈ Im Q_{θ}, there exists a unique x̄ = (x_{n})_{n∈ℤ} ∈ Im Q_{θ} such that (see (4.1))

{Φ(φ^{n−k}θ, k)x_{n−k}}_{n∈ℤ} = π_{θ}(T^{k})x̄ = ȳ.

The fact ȳ ∈ Im Q_{θ} means precisely that y_{n} ∈ Im Q(φ^{n}θ) for every n ∈ ℤ. So fix y ∈ Im Q(φ^{k}θ) and define ȳ by y_{n} = y for n = k, and y_{n} = 0, for n ≠ k. There exists a unique x̄ ∈ Im Q_{θ} such that

Φ(φ^{n−k}θ, k)x_{n−k} = y_{n}, n ∈ ℤ.

In particular, for n = k, Φ(θ, n)x_{0} = y. This shows that Φ(θ, n) is invertible from Im Q(θ) to Im Q(φ^{n}θ) for all θ ∈ Θ, n ∈ ℤ.

To see that Φ(θ, t) is invertible for all t ∈ ℝ, fix t and choose n ∈ ℤ such that t ∈ [n, n + 1). Since Φ(θ, n) is invertible, the identity

Φ(θ, t) = Φ(φ^{n}θ, t − n)Φ(θ, n)

shows that it suffices to prove Φ(φ^{n}θ, t − n) is invertible. But the identities

\begin{aligned}Φ(φ^{n}θ, 1) & = Φ(φ^{t}θ, n + 1 − t)Φ(φ^{n}θ, t − n) \\ Φ(φ^{t−1}θ, 1) & = Φ(φ^{n}θ, t − n)Φ(φ^{t−1}θ, n − (t − 1)) \end{aligned}

show, respectively, that Φ(φ^{n}θ, t − n) has a left and a right inverse. Hence Φ(θ, t) is invertible from Im Q(θ) to Im Q(φ^{t}θ).

Part b) for t ∈ ℤ_{+} follows from (4.1) and the fact that π_{θ}(T)P = π_{θ}(T)P. As in [25, Prop. 3.10], the statement is seen to hold for t ∈ ℝ_{+}.

Finally, since σ(T^{t}) ∩ 𝕋 = ∅, there exist M, β > 0 such that

sup_{t∈ℝ}∥Φ(θ, t)P (θ)∥_{L(X)} = ∥PT^{t}P∥_{B(C_{0}(Θ,X))} ≤ Me^{−βt}.

The first inequality in Definition 4.1 c) follows. The second inequality is shown similarly. □

The Proof of iii) ⇒ i) is trivial. This completes the proof of Theorem 4.1.

Let us make the following observation concerning the dynamical spectrum, Σ (see the Introduction for the definition). As a result of Theorem 4.1 and Corollary 4.5, the dynamical spectrum Σ coincides

32 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

with ln |σ(T) \ {0}|. An application of Theorem 3.3 gives the formula (1.3) in the introduction. Also, the spectral subbundles for φ̂^{t} are determined by the spectral projections for T.

5. Consequences

In this section we formulate a variety of consequences which follow, almost immediately, from the results of the previous sections.

5.1. Invertibility of Γ. An immediate consequence of the Theorem 3.3 and Theorem 4.1 ( see i) ⇔ iii)) is the fact that a LSPF has exponential dichotomy if and only if the spectrum of the generator Γ of the corresponding evolutionary semigroup (2.2) satisfies σ(Γ) ∩ iℝ = ∅. Since σ(Γ) is invariant with respect to translations along iℝ, we conclude:

Corollary 5.1. The LSPF φ̂^{t} has exponential dichotomy if and only if Γ is invertible.

Now recall that in the norm continuous compact setting, the generator of the evolutionary semigroup is given by (2.5). Consequently, the LSPF generated by the cocycle in Example 2.2 is hyperbolic if and only if the equation

\frac{d}{dt}f ∘ φ^{t}(θ)|_{t=0} − A(θ)f(θ) = g(θ)

has a unique solution f for every g ∈ C_{0}(Θ, X).

Applying Corollary 5.1 to Example 2.8, we note that an evolutionary family {U(τ, s)}, τ ≥ s has exponential dichotomy if and only if Γ is invertible on C_{0}(ℝ, X) (cf. [22, 23, 24]). In the special case where A(τ) ≡ A_{0} generates a C_{0} semigroup {e^{tA_{0}}}_{t≥0} on X (here, U(τ, s) = e^{(τ−s)A_{0}}), this gives a result from [41]: a C_{0} semigroup {e^{tA_{0}}}_{t≥0} is hyperbolic if and only if the equation f′ − A_{0}f = g has a unique solution for every g ∈ C_{0}(ℝ, X).

On the other hand, for the norm continuous line setting (Example 2.7), we conclude that (2.10) has exponential dichotomy if and only if Γ = −d/dt + A(·) is invertible [3, 37], or, equivalently, the equation f′(t) − A(t)f(t) = g(t), t ∈ ℝ, has a unique solution for every g ∈ C_{0}(ℝ, X) (see [11, 31]).

EVOLUTIONARY SEMIGROUPS AND  DICHOTOMY OF LSPF  33

5.2. Roughness of the dichotomy. In this subsection we give a very short proof of the facts that the exponential dichotomy persists under small perturbations. Theorem 5.2. Assume that the LSPF φ̂_{1}^{t} over the flow φ^{t} generated by a cocycle Φ_{1}(θ, t) has exponential dichotomy. Then there exists ϵ > 0 such that for every cocycle Φ_{2}(θ, t) satisfying

sup_{θ∈Θ} ∥Φ_{1}(θ, 1) − Φ_{2}(θ, 1)∥_{L(X)} < ϵ,  (5.1)

the LSPF φ̂_{2}^{t} over φ^{t} generated by Φ_{2}(θ, t) also has the exponential dichotomy.

Proof. By Theorem 4.1, i) ⇔ iii), the operator T_{1}, (T_{1}f)(θ) = Φ_{1}(φ^{−1}θ, 1)f(φ^{−1}θ), is hyperbolic in C_{0}(Θ, X). Note that for (T_{2}f)(θ) = Φ_{2}(φ^{−1}θ, 1)f(φ^{−1}θ),

∥T_{1} − T_{2}∥_{B(C_{0}(Θ,X))} = sup_{∥f∥_{C_{0}(Θ,X)}=1} ∥[Φ_{1}(φ^{−1}· , 1) − Φ_{2}(φ^{−1}· , 1)] f(φ^{−1}·)∥_{C_{0}(Θ,X)}

≤ sup_{θ∈Θ} ∥Φ_{1}(θ, 1) − Φ_{2}(θ, 1)∥_{L(X)}.

For sufficiently small ϵ > 0, (5.1) implies σ(T_{2}) ∩ 𝕋 = ∅. □

¿From Corollary 5.1 we derive the following. Corollary 5.3. Let φ̂_{1}^{t} and φ̂_{2}^{t} be two LSPFs over φ^{t}, and let Γ_{1} and Γ_{2} denote the respective generators of the corresponding evolutionary semigroups (2.2). If φ̂_{1}^{t} has exponential dichotomy, then there exists ϵ > 0, ϵ = ϵ(φ̂_{1}^{t}) such that ∥Γ_{1}−Γ_{2}∥ < ϵ implies φ̂_{2}^{t} also has exponential dichotomy.

Since the “typical” generator Γ for a variational equation (2.4) has the form (2.5), this can be used when the unbounded operators A(θ) are perturbed by bounded operators such that Γ_{1} − Γ_{2} is bounded.

For the strongly continuous setting, we have the following roughness result (cf. [38]) for the variational equation in Example 2.5. Recall that in that example the generator of the evolutionary semigroup is identified in (2.7). Corollary 5.4. Let A_{0} be a generator of a C_{0} semigroup {e^{tA_{0}}}_{t≥0} on X, and assume A_{1} and A_{2} : Θ → L_{s}(X) be bounded and continuous. Consider the equations:

\frac{dx}{dt} = A_{0}x(t)+A_{1}(φ^{t}θ)x(t), \frac{dx}{dt} = A_{0}x(t)+A_{2}(φ^{t}θ)x(t), θ ∈ Θ, t ∈ ℝ

34 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

(see (2.4)–(2.6)). Assume that the LSPF φ̂_{1}^{t} over φ^{t} generated by the first equation has exponential dichotomy. Then there exists ϵ > 0, ϵ = ϵ(φ̂_{1}^{t}) such that ∥A_{1}(θ) − A_{2}(θ)∥ < ϵ implies the LSPF φ̂_{2}^{t} over φ^{t} generated by the second equation has exponential dichotomy, provided the both cocycles are strongly continuous.

This gives an alternate way of addressing the situation of Theorem 5.2 in [8].

5.3. Green’s function. We describe the existence of exponential dichotomy for φ̂^{t} in terms of the existence and uniqueness of a Green’s function. Let P: Θ → L_{s}(X) be a bounded continuous projectionvalued function that satisfies the following properties for θ ∈ Θ and t > 0:

a) Φ(θ, t)P (θ) = P (φ^{t}θ)Φ(θ, t);

b) Φ(θ, t) is invertible as an operator from Im Q(θ) to Im Q(φ^{t}θ). Define

G(θ, t) = \begin{cases}Φ_{P} (θ, t), & t > 0 \\ Φ_{Q}(θ, t), & t < 0. \end{cases}  (5.2)

Here we have denoted:

Φ_{P} (θ, t) = P (φ^{t}θ)Φ(θ, t)P (θ) for t > 0
Φ_{Q}(θ, t) = [Q(φ^{−t}θ)Φ(θ, −t)Q(θ)]^{−1} for t < 0,

and used a) and b). Definition 5.1. We say that the LSPF φ̂^{t} has a Green’s function if there exists a bounded continuous projection P: Θ → L_{s}(X), such that for G from (5.2) the operator

(Ĝf)(θ) = ∫_{−∞}^{∞} G(φ^{−t}θ, t)f(φ^{−t}θ) dt, θ ∈ Θ,  (5.3)

is bounded on C_{0}(Θ, X).

The following result generalizes [49] (see also [21]). Theorem 5.5. The LSPF φ̂^{t} has exponential dichotomy if and only if the Green’s function exists and is unique.

Proof. For every P (·) satisfying a)-b), define P ∈ B(C_{0}(Θ, X)) by Pf(θ) = P (θ)f(θ), and set Q = I − P. Then P commutes with T^{t}, and T_{Q}^{t} ≡ QT^{t}Q is invertible in Im Q. We note that (5.3) can be rewritten as Ĝ = −G̃, where

G̃f = ∫_{0}^{∞} T_{Q}^{−t}f dt − ∫_{0}^{∞} T_{Q}^{t}f dt.  (5.4)

If Γ denotes the generator of T^{t}, then the Spectral Mapping Theorem, 3.3, shows that Γ^{−1} ∈ B(C_{0}(Θ, X)) if and only if T^{t} is hyperbolic.

Assume the Green’s function, G, exists and is unique. Then G̃ is bounded. By [24, Lemma 4.2] then Γ is invertible and Γ^{−1} = G̃. Since T^{t} is hyperbolic, i) ⇒ iii) of Theorem 4.1 shows that φ̂^{t} has exponential dichotomy.

Conversely, assume φ̂^{t} has exponential dichotomy. Then T^{t} is hyperbolic with Riesz projection P = P (·). By [2] this projection is unique. The standard norm-estimates in (5.4) shows that G̃ and Ĝ are bounded. □

5.4. Pointwise dichotomy. In this subsection we consider the interrelation between “global” and “pointwise” dichotomies. Let 𝕂 denote either ℝ or ℤ. Fix a point θ_{0} ∈ Θ. We define an exponential dichotomy of φ̂^{t} over the orbit through θ_{0} as follows.

Definition 5.2. The LSPF φ̂^{t} has exponential dichotomy over 𝕂 at θ_{0} if for all τ ∈ 𝕂, there exists a projection P (φ^{τ}θ_{0}) ∈ L(X) such that

a) P (φ^{τ}θ_{0})Φ(θ_{0}, τ) = Φ(θ_{0}, τ)P (θ_{0});

b) Φ_{Q}(φ^{τ}θ_{0}, t) is invertible from Im Q(φ^{τ}θ_{0}) to Im Q(φ^{τ+t}θ_{0}) for all t > 0 and τ ∈ 𝕂;

c) there exist positive constants M = M(θ_{0}), β = β(θ_{0}) such that for all τ ∈ ℝ and t > 0:

∥Φ_{P} (φ^{τ}θ_{0}, t)∥ ≤ Me^{−βt}, ∥ [Φ_{Q}(φ^{τ}θ_{0}, t)]^{−1} ∥ ≤ Me^{−βt}.

For 𝕂 = ℝ we require, in addition, that τ ↦ P (φ^{τ}θ_{0}) is a (strongly) continuous function from ℝ to L_{s}(X).

For invertible-valued cocycles Φ: Θ × 𝕂 → L(X), Definition 5.2 is equivalent to the classical definition of exponential dichotomy of a LSPF at a point: There exists a projection P and constants M, β, such that ∥Φ(θ_{0}, s′)P Φ^{−1}(θ_{0}, s)∥ ≤ Me^{−β(s′−s)} for s′ ≥ s, and ∥Φ(θ_{0}, s′)QΦ^{−1}(θ_{0}, s)∥ ≤ Me^{−β(s−s′)} for s ≥ s′ (see, e.g., [46]).

36 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

The following assertion has appeared previously. For the sake of completeness, we include a brief proof that is consistent with the approach of the present paper (see also [8], [16, Thm. 7.6.5], and [25, Thm. 3.22]).

Lemma 5.6. The LSPF φ̂^{t} has exponential dichotomy over ℤ at θ_{0} ∈ Θ if and only if π_{θ_{0}}(T) is hyperbolic in l_{∞}(ℤ; X).

Proof. Indeed, the spectral radius r of π_{θ_{0}}(T)P_{θ_{0}} is given by the formula

r = lim_{k→∞} (sup_{n∈ℤ}∥Φ(φ^{n}θ_{0}, k)P (φ^{n}θ_{0})∥)^{1/k} ,

and the exponential dichotomy of φ̂^{t} at θ_{0} implies the hyperbolicity of π_{θ_{0}}(T) with Riesz projection

P_{θ_{0}} = diag{P_{n}}_{n∈ℤ}  (5.5)

for P_{n} := P (φ^{n}θ_{0}).

Conversely, assume π_{θ_{0}}(T) ∩ 𝕋 = ∅. Then for all λ ∈ 𝕋, [λI − π_{θ_{0}}(T)]^{−1} ∈ 𝔇. Therefore, as in Corollary 4.5, the Riesz projection P_{θ_{0}} corresponding to σ(π_{θ_{0}}(T) ∩ 𝔻 is an element of 𝔇. Proceeding as in [21, Lemma 3], one can show that P_{θ_{0}} ∈ ℭ. I.e., there exist operators P_{n} in L(X) such that P_{θ_{0}} = diag{P_{n}}_{n∈ℤ}. Now define P (φ^{n}θ_{0}) := P_{n} for each n ∈ ℤ. This defines projections in L(X), which, by the fact that π_{θ_{0}}(T)P_{θ_{0}} = P_{θ_{0}}π_{θ_{0}}(T), are seen to satisfy part a) of Definition 5.2:

P (φ^{τ}θ_{0})Φ(θ_{0}, τ) = Φ(θ_{0}, τ)P (θ_{0}), τ ∈ ℤ

(use (4.1)). Also, as in the proof of Lemma 4.7, Φ_{Q}(θ_{0}, k): Im Q(θ_{0}) → Im Q(φ^{k}θ_{0}) is invertible for all θ ∈ Θ, k ∈ ℤ. In particular, Φ_{Q}(φ^{n}θ_{0}, k): Im Q(φ^{n}θ_{0}) → Im Q(φ^{n+k}θ_{0}) is invertible for all n, k ∈ ℤ.

The estimates in Definition 5.2 c) are also verified as in Lemma 4.7. □

Combining this with Theorem 4.1 gives the following fact.

Corollary 5.7. The LSPF φ̂^{t} has exponential dichotomy on Θ if and only if it has exponential dichotomy at every θ ∈ Θ and ∥[π_{θ}(T) − λI]^{−1}∥ ≤ C, θ ∈ Θ, λ ∈ 𝕋.

EVOLUTIONARY SEMIGROUPS AND  DICHOTOMY OF LSPF  37

5.5. Evolutionary semigroups along trajectories. Next we relate the exponential dichotomy of φ̂^{t} to an associated evolutionary semigroup defined on C_{0}(ℝ, X) along each trajectory of the flow (cf. [21, 25]). Fix θ ∈ Θ, and consider the semigroup {Π_{θ}^{t}}_{t≥0} defined by

(Π_{θ}^{t}f)(s) = Φ(φ^{s−t}θ, t)f(s − t), s ∈ ℝ, t ≥ 0, f ∈ C_{0}(ℝ, X).

Let L_{θ} denote the generator. The semigroup {Π_{θ}^{t}}_{t≥0} is, in fact, an evolutionary semigroup in the sense of the strongly continuous line setting. Indeed, for s ≥ τ, set U(s, τ) = Φ(φ^{τ}θ, s − τ). Then U(s, s) = I, and for s ≥ r ≥ τ,

U(s, τ) = Φ(φ^{τ}θ, s − t) = Φ(φ^{r−τ}(φ^{τ}θ), s − r)Φ(φ^{τ}θ, r − t)
= Φ(φ^{r}θ, s − r)Φ(φ^{τ}θ, r − τ)
= U(s, r)U(r, τ).

Hence, {U(s, τ)}_{s≥τ} is an evolutionary family on X, and (Π_{θ}^{t}f)(s) = U(s, s − t)f(s − t). As shown in [22, 23] (see also [44, 45]), any such evolutionary semigroup has the following properties: Lemma 5.8. σ(Π_{θ}^{t}) is invariant under rotations centered at origin, σ(L_{θ}) is invariant under translation along iℝ, and the spectral mapping theorem holds:

σ(Π_{θ}^{t})\{0} = e^{tσ(L_{θ})}, t > 0.

Further, when σ(Π_{θ}^{1}) ∩ 𝕋 = ∅, the corresponding Riesz projection P is an operator of multiplication: (Pf)(s) = P (s)f(s), for some bounded, continuous projection-valued P : ℝ → L_{s}(X).

Theorem 4.1 and Lemma 5.6 give the following result. Theorem 5.9. The following are equivalent

i) The LSPF φ̂^{t} has exponential dichotomy;

ii) σ(Π_{θ}^{1}) ∩ 𝕋 = ∅ for all θ ∈ Θ, and there exists C > 0 such that

∥ [Π_{θ}^{1} − λI]^{−1} ∥_{B(C_{0}(ℝ;X))} ≤ C, θ ∈ Θ, λ ∈ 𝕋;

iii) σ(L_{θ}) ∩ iℝ = ∅ for all θ ∈ Θ, and there exists C > 0 such that

∥ [iξ − L_{θ}]^{−1} ∥_{B(C_{0}(ℝ;X))} ≤ C, θ ∈ Θ, ξ ∈ ℝ.

As in Theorem 4.1 (see also [LMS1]), one can replace the inequalities in ii) and iii) by the inequalities

∥[Π_{θ}^{1} − I]^{−1}∥_{B(C_{0}(ℝ;X))} ≤ C, and ∥L_{θ}^{−1}∥_{B(C_{0}(ℝ;X))} ≤ C, θ ∈ Θ,

38 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

respectively. We note that the operators L_{θ} are differential operators of the first order, and i) ⇔ iii) reduces the problem of the existence of exponential dichotomy on Θ to the problem of invertibility of these operators.

Proof. Lemma 5.8 proves the equivalence of iii) and ii). To prove i) ⇔ ii), we introduce operators on the space

C_{0}(ℝ × Θ, X) = C_{0}(Θ, C_{0}(ℝ, X)) = C_{0}(ℝ, C_{0}(Θ, X))

defined as

\begin{aligned}(Πh)(s, θ) & = Φ(φ^{s−1}θ, 1)h(s − 1, θ), \\ (T̂h)(s, θ) & = Φ(φ^{−1}θ, 1)h(s − 1, φ^{−1}θ), \\ (Jh)(s, θ) & = h(s, φ^{s}θ), (J^{−1}h)(s, θ) = h(s, φ^{−s}θ), \end{aligned}

for s ∈ ℝ, θ ∈ Θ, h ∈ C_{0}(ℝ × Θ, X). These operators satisfy

J^{−1}ΠJ = T̂.  (5.6)

Indeed, for r_{1}(s, θ) = h(s, φ^{s}θ) one has (Πr_{1})(s, θ) = Φ(φ^{s−1}θ, 1)h(s − 1, φ^{s−1}θ), and (J^{−1}ΠJh)(s, θ) = Φ(φ^{s−1}(φ^{−s}θ), 1)h(s−1, φ^{s−1}(φ^{−s}θ) = (T̂h)(s, θ).

Next, note that for a function F: Θ → C_{0}(ℝ, X), the operator Π acts as the multiplication by Π_{θ}^{1}: (ΠF)(θ) = Π_{θ}^{1}F (θ). Hence, for λ ∈ 𝕋, the operator λ − Π of multiplication by λ − Π_{θ}^{1} is invertible on C_{0}(Θ, C_{0}(ℝ, X)) if and only if λ − Π_{θ}^{1} is invertible on C_{0}(ℝ, X) for each θ ∈ Θ, and ∥(λ − Π_{θ}^{1})^{−1}∥ ≤ C for some C > 0. This means, that the statement σ(Π) ∩ 𝕋 = ∅ is equivalent to ii).

By Theorem 4.1, i) is equivalent to the statement σ(T) ∩ 𝕋 = ∅ on C_{0}(Θ, X). From [22] (Theorem 2.5, (1) ⇔ (2)), we conclude that σ(T)∩ 𝕋 = ∅ on C_{0}(Θ, X) is equivalent to σ(T̂) ∩ 𝕋 = ∅ on C_{0}(ℝ, C_{0}(Θ, X)). Indeed, for f: ℝ → C_{0}(Θ, X), T̂ acts as (T̂f)(s) = T f(s − 1), and this is exactly the case considered in Theorem 2.5 of [22]. Hence, i) is equivalent to σ(T̂) ∩ 𝕋 = ∅. Equation (5.6) shows that σ(Π) = σ(T̂), and hence i) ⇔ ii). □

5.6. Open Problems and Concluding Remarks. Remark 1. An open problem is to prove analogues of Theorems 3.3 and 4.1 in the L_{p}-setting: Let μ be a Borel measure on Θ, positive on open sets, and quasi-invariant with respect to the flow φ^{t}. The semigroup (2.2) could be replaced by

(T^{t}f)(θ) = (\frac{dμ ∘ φ^{t}}{dμ} (θ))^{1/p} Φ(φ^{−t}θ, t)f(φ^{−t}θ), f ∈ L_{p}(Θ, μ, X).

The proof of Theorem 3.3 would require minor changes. However, the proof of Lemma 4.3 should be modified essentially. Theorem 4.1 has been proven for the strongly continuous line setting for L_{p} [22, 24]; in this setting, the statement i) ⇔ iii) is proved in [42] using an interesting alternative C_{0}-semigroup approach. Remark 2. It would be interesting to determine under which conditions on the cocycle Φ and in which sense (in strongly continuous locally compact setting) the generator Γ is given by formula (2.5) (see [34, 35] for the line setting). Remark 3. We conjecture that the condition (4.2) in Theorem 4.1 and the corresponding inequalities in Corollary 5.7 and Theorem 5.9 are redundant, at least for the norm continuous compact setting. This means that the LSPF is exponentially dichotomic on Θ if and only if it is exponentially dichotomic at every θ ∈ Θ. This conjecture is true for finite dimensional X [47, Lemma 2A] and for the norm continuous compact setting where X is a Hilbert space [25]. The corresponding algebraic question here is whether the set of representations {π_{θ}}_{θ∈Θ} of the algebra 𝔅 is sufficient, that is, whether b ∈ 𝔅 is invertible if and only if π_{θ}(b) is invertible for all θ ∈ Θ. Remark 4. A problem related to Remark 3 is to consider, instead of π_{θ}(T), weighted shift operators πθ̄(T) along ϵ-trajectories θ̄. Recall, that a sequence θ̄ = {θ_{n}}_{n∈ℤ} is called an ϵ-trajectory for φ = φ^{1} if dist(φθ_{n}, θ_{n+1}) ≤ ϵ for n ∈ ℤ. The operator πθ̄(T) can be defined on l_{∞}(ℤ, X) as πθ̄(T) = diag{Φ(θ_{n−1}, 1)}_{n∈ℤ}S. We suspect that the LSPF has exponential dichotomy over Θ if and only if πθ̄(T) is hyperbolic for all ϵ-trajectories with sufficiently small ϵ. Remark 5. An area open for investigation is the case when Φ is a (semi)cocycle over a semiflow {φ^{t}}_{t∈ℝ_{+}}. For the line setting, this corresponds to the dichotomy on the semiaxis. Remark 6. We showed in this paper that exponential dichotomy persists under small perturbations of the cocycle. A theorem by R. Sacker

40 Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

and G. Sell [47, Thm. 6] says that dichotomy persists under “small” perturbations of a φ^{t}-invariant compact subset Θ_{0} ⊂ Θ. The proof in [47] is essentially finite-dimensional. It would be of interest to know whether the theorem holds in the strongly continuous setting. We note that Remark 4 helps to prove the theorem for the uniformly continuous setting when X is a Hilbert space (in preparation). Remark 7. We note that Lyapunov numbers (see [8, 20, 47]) for the cocycle Φ belong to the dynamical spectrum Σ = σ(Γ) ∩ ℝ. As proven in [20] for the finite dimensional setting and in [25] for the uniformly continuous setting on a Hilbert space X, the boundaries of Σ can be computed via the exact Lyapunov-Oseledets exponents given by the multiplicative ergodic theorem. This theorem is now available for Banach spaces [29]. A natural question then is to characterize the boundaries of Σ for the compact-valued cocycles on a Banach space. Remark 8. Another natural question is to relax our main assumption that the aperiodic trajectories of φ^{t} are dense in Θ. Without this assumption the Spectral Mapping Theorem does not hold (see [4] and [25]). Let p(θ) = inf{t : φ^{t}θ = θ} denote the prime period of θ ∈ Θ, and set

p_{0}(θ) := inf_{U} sup_{y∈U} p(y),

where U denotes an open set containing θ. Let H(S) denote the union of the circles, centered at origin, intersecting a set S ⊂ ℂ.

Assume p_{0}(θ) ≥ c > 0 for all θ ∈ Θ and some c. We conjecture that the following Annular Hull Theorem (see [4]) is valid:

exp tσ(Γ) ⊂ σ(T^{t}) \ {0} ⊂ H(exp tσ(Γ)).

There are infinite-dimensional counterexamples (similar to one in [32]) showing the theorem fails without the assumption p_{0}(θ) ≥ c > 0.

To prove this conjecture it might be helpful to consider the following function:

y(θ) = ∫_{0}^{p_{0}(θ)}ρ(t/p_{0}(θ))(T^{t}w)(θ) + (1 − ρ(t/p_{0}(θ)))(T^{t+p_{0}(θ)}w)(θ)dt,

instead of (3.11) in the proof of the Spectral Mapping Theorem. The function ρ : [0, 1] → [0, 1] here (see [23, p. 46]) is a smooth function such that that ρ(τ) = 0 for τ ∈ [0, 1/3] and ρ(τ) = 1 for τ ∈ [2/3, 1].

6. Acknowledgments

Y. Latushkin is grateful to Professor Shui-Nee Chow, Professor Jack Hale, and Professor Yingfei Yi for the suggestions, questions, and discussions during his visit to Georgia Tech in 1994, and to Hugo Leiva for helpful discussions on Remarks 3 and 6.

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Department of Mathematics, University of Missouri, Columbia, MO 65211

  E-mail address: mathyl@mizzou1.missouri.edu

Department of Mathematics, University of Missouri, Columbia, MO 65211

  E-mail address: stephen@mont.cs.missouri.edu

Department of Mathematics, University of Missouri, Rolla, MO 65401

  E-mail address: randolph@umrvmb.umr.edu