36
EVOLUTION SEMIGROUPS, TRANSLATION ALGEBRAS, AND EXPONENTIAL DICHOTOMY OF COCYCLES YURI LATUSHKIN AND ROLAND SCHNAUBELT Abstract. We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space Θ acting on a Banach space X. Our main tool is the associated evolution semigroup on C 0 (Θ; X). We prove that the cocycle has exponential dichotomy if and only if the evolution semi- group is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 (Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underly- ing compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation. 1. Introduction Exponential dichotomy of cocycles and nonautonomous Cauchy problems is a classi- cal and well-studied subject, see [13, 14, 17, 18, 30, 35, 42, 43] and the literature cited therein. Recently, important contributions were made in the infinite dimensional case, see [10, 11, 12, 28, 44], and in the applications, [46, 47]. In this paper we continue the investigation, begun in [21, 25], of the dichotomy of cocycles by means of the so-called evolution semigroup. Let {ϕ t } tR be a continuous flow on a locally compact metric space Θ and let {Φ t } tR + be an exponentially bounded, strongly continuous cocycle over {ϕ t } with values in the set L(X ) of bounded linear operators on a Banach space X . Cocycles arise, for instance, as solution operators for variational equations ˙ u(t)= A(ϕ t θ)u(t) with possibly unbounded operators A(θ), see e.g. [21, 42, 44]. On the space F = C 0 (Θ; X ) of continuous functions f X vanishing at infinity endowed with the sup-norm, we 1991 Mathematics Subject Classification. 47D06, 34G10, 34C35, 34D10, 47A10. Key words and phrases. Dichotomy, hyperbolicity, cocycle, evolution semigroup, spectral mapping theorem, translation algebras. YL: Supported by the NSF grant DMS-9622105 and by the Research Board of the University of Missouri. RS: Support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. 1

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EVOLUTION SEMIGROUPS, TRANSLATION ALGEBRAS, ANDEXPONENTIAL DICHOTOMY OF COCYCLES

YURI LATUSHKIN AND ROLAND SCHNAUBELT

Abstract. We study the exponential dichotomy of an exponentially bounded, strongly

continuous cocycle over a continuous flow on a locally compact metric space Θ acting on

a Banach space X. Our main tool is the associated evolution semigroup on C0(Θ;X).

We prove that the cocycle has exponential dichotomy if and only if the evolution semi-

group is hyperbolic if and only if the imaginary axis is contained in the resolvent set of

the generator of the evolution semigroup. To show the latter equivalence, we establish

the spectral mapping/annular hull theorem for the evolution semigroup. In addition,

dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift

operators defined on c0(Z;X). Here we develop Banach algebra techniques and study

weighted translation algebras that contain the evolution operators. These results imply

that dichotomy persists under small perturbations of the cocycle and of the underly-

ing compact metric space. Also, exponential dichotomy follows from pointwise discrete

dichotomies with uniform constants. Finally, we extend to our situation the classical

Perron theorem which says that dichotomy is equivalent to the existence and uniqueness

of bounded, continuous, mild solutions to the inhomogeneous equation.

1. Introduction

Exponential dichotomy of cocycles and nonautonomous Cauchy problems is a classi-

cal and well-studied subject, see [13, 14, 17, 18, 30, 35, 42, 43] and the literature cited

therein. Recently, important contributions were made in the infinite dimensional case,

see [10, 11, 12, 28, 44], and in the applications, [46, 47]. In this paper we continue the

investigation, begun in [21, 25], of the dichotomy of cocycles by means of the so-called

evolution semigroup.

Let ϕtt∈R be a continuous flow on a locally compact metric space Θ and let Φtt∈R+

be an exponentially bounded, strongly continuous cocycle over ϕt with values in the

set L(X) of bounded linear operators on a Banach space X. Cocycles arise, for instance,

as solution operators for variational equations

u(t) = A(ϕtθ)u(t)

with possibly unbounded operators A(θ), see e.g. [21, 42, 44]. On the space F = C0(Θ;X)

of continuous functions f : Θ→ X vanishing at infinity endowed with the sup-norm, we

1991 Mathematics Subject Classification. 47D06, 34G10, 34C35, 34D10, 47A10.Key words and phrases. Dichotomy, hyperbolicity, cocycle, evolution semigroup, spectral mapping

theorem, translation algebras.YL: Supported by the NSF grant DMS-9622105 and by the Research Board of the University of

Missouri.

RS: Support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.1

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define the evolution semigroup Ett∈R+ by

(Etf)(θ) = Φt(ϕ−tθ)f(ϕ−tθ) for θ ∈ Θ, t ≥ 0, f ∈ F . (1.1)

The semigroup Et is strongly continuous on C0(Θ;X). Its generator is denoted by Γ.

We briefly review the history of the subject before we outline the results and methods

presented in the current paper. Starting with the pioneering paper [31] by J. Mather,

spectral properties of evolution semigroups have been used to characterize exponential

dichotomy of the underlying cocycle. In fact, for the finite dimensional case dimX <∞and compact Θ, the following Dichotomy Theorem is contained in [31]:

Φt has exponential dichotomy on Θ if and only if Et is hyperbolic;

that is, σ(E1) ∩ T = ∅ for the spectrum σ(E1) and the unit circle T. Moreover, in the

finite dimensional, compact setting it was proved in [7, 8, 9, 19] that the semigroup Etsatisfies the spectral mapping theorem

σ(Et)\0 = exp(tσ(Γ)) for t ≥ 0 (1.2)

provided that the flow ϕt is aperiodic (i.e., Intθ ∈ Θ : ϕT θ = θ for some T > 0 = ∅),and the annular hull theorem

(σ(Et)\0) · T = exp(tσ(Γ)) · T for t ≥ 0 (1.3)

with no restrictions on the flow ϕt. We point out that the spectral mapping theorem

does not hold without the assumption of the aperiodicity of the flow, see [9, Ex. 2.3].

As a consequence, the dichotomy of the cocycle is equivalent to the spectral condition

σ(Γ) ∩ iR = ∅. An essential step in the proof of these results was a proposition from [29]

known as the Mane Lemma, see also [7, 8, 9]. It says that 1 belongs to the approximate

point spectrum σap(E1) if and only if there exists a point θ0 ∈ Θ, called the Mane point,

such that sup|Φt(θ0)x0| : t ∈ R is finite for a nonzero vector x0. We stress that,

in general, the Mane Lemma does not hold in the infinite dimensional or/and locally

compact setting due to the noncompactness of the unit sphere in X or/and Θ and to the

fact that Φt is defined only for t ≥ 0.

The operator E1 =: E can be expressed as a weighted translation operator E = aV on

F , where a(θ) = Φ1(ϕ−1θ) and (V f)(θ) = f(ϕ−1θ). An important C∗-algebra technique

for the study of weighted translation operators E on the space L2(Θ;X) was developed

in [1, 2]. In particular, the hyperbolicity of the operator E was related to that of a family

of weighted shift operators πθ(E) acting on a space of X-valued sequences x = (xn)n∈Zby the rule

πθ(E) : (xn)n∈Z 7→ (Φ1(ϕn−1θ)xn−1)n∈Z = diag((a(ϕnθ))n∈Z S (xn)n∈Z (1.4)

for θ ∈ Θ, where S(xn)n∈Z = (xn−1)n∈Z.

For compact Θ and infinite dimensionalX, the evolution semigroup (1.1) was considered

in [25] for eventually norm-continuous in θ cocycles, cf. Section 3.3, and in [40] for strongly

continuous cocycles with invertible values. The proof of the Dichotomy Theorem from

[40] requires just a straightforward modification to work in our setting.

The special case of the translation flow ϕtθ = θ + t on Θ = R is well understood

by now, see [4, 5, 20, 22, 34, 36, 37, 39]. Here, the cocycle is given by Φt(θ) = U(θ +

t, θ) for a strongly continuous evolution family U(t, s)t≥s on a Banach space X which2

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can be thought as solution operator for the evolution equation u(t) = A(t)u(t) with

unbounded operators A(t). We refer to [45] for applications of the spectral theory of

evolution semigroups to parabolic evolution equations.

In [21] and [24] the general locally compact and infinite dimensional setting was inves-

tigated under the additional and restrictive assumption that the underlying flow ϕt is

aperiodic. In particular, the spectral mapping theorem (1.2) was established in [21] for

F = C0(Θ;X) and in [24] for F = Lp(Θ;X). Also, the following Discrete Dichotomy

Theorem was proved in [21]: The semigroup Et is hyperbolic on C0(Θ;X) if and only

if σ(πθ(E))∩T = ∅ for each θ ∈ Θ on the space c0(Z;X) of X-valued sequences vanishing

at infinity and

supz∈T,θ∈Θ

‖[z − πθ(E)]−1‖L(c0(Z;X)) <∞.

Via the Dichotomy Theorem, the Discrete Dichotomy Theorem relates “global” dichotomy

of Φt on Θ and “pointwise” dichotomies at θ ∈ Θ, cf. [10, 18].

The proof of the Discrete Dichotomy Theorem in [21] required a further development of

the algebraic techniques from [2, 20, 22, 25], see also [16]. Namely, the weighted translation

operator E was “immersed” in the algebra B of operators b ∈ L(C0(Θ;X)) of the form

b =∞∑

k=−∞

akVk with ‖b‖1 :=

∞∑k=−∞

‖ak‖L(C0(Θ;X)) <∞,

where ak are the multiplication operators induced by strongly continuous, bounded func-

tions ak : Θ → L(X). The map πθ : E 7→ πθ(E) was extended to a representation of B

in L(c0(Z;X)). We stress that without the aperiodicity assumption on ϕt the algebra

B cannot even be defined. Thus, a generalization to arbitrary flows requires a radical

modification of the method.

In the present paper, we consider evolution semigroups Et and cocycles Φt without

any additional assumptions on ϕt. We recall necessary background material and prove

the Dichotomy Theorem for strongly continuous cocycles in Section 2. The persistence of

dichotomy under small perturbations of the cocycle follows immediately from this result,

cf. [10].

A modification of the Banach algebra techniques mentioned above is developed in Sec-

tion 3. Here, we replace the algebra B by a certain convolution algebra B. This allows

to prove the Discrete Dichotomy Theorem. In addition, for θ ∈ Θ, we introduce a family

of strongly continuous semigroups Πtθt∈R+ on C0(R;X) by setting

(Πtθh)(s) = Φt(ϕs−tθ)h(s− t) for s ∈ R, t ≥ 0, h ∈ C0(R;X).

Each of the semigroups Πtθ is an evolution semigroup in the sense of [20, 37]. If Φt(θ) is

the solution operator for the variational equation u(t) = A(ϕtθ)u(t), then the generator

Γθ of Πtθ is the closure of the operator h 7→

[− dds

+ A(ϕ·θ)]h, see [21]. We give an

analogue of the Discrete Dichotomy Theorem for Πtθ. As an application, we derive

for eventually norm-continuous in θ cocycles an infinite dimensional generalization of the

Sacker-Sell Perturbation Theorem [43, Thm. 6] on the semicontinuity of the Sacker-Sell

dynamical spectrum as a function of the underlying compact set Θ.

In Section 4, we prove the spectral mapping theorem for aperiodic flows and the an-

nular hull theorem provided that the function of prime periods is strictly separated from3

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zero. (In fact, we assume somewhat weaker properties of the flow.) Note that the last

assumption is needed for the Annular Hull Theorem 1.3 if dimX =∞. Counterexamples

are furnished by the identical flow on a singleton and the cocycle given by a semigroup

violating the annular hull theorem, see [33, 34] for such semigroups. We stress that the

annular hull theorem does not hold even for the semigroup related to a first-order pertur-

bation of the two-dimensional wave equation, see [41]. In the proof of the annular hull and

spectral mapping theorems for the evolution semigroups we give an explicit construction

of approximate eigenfunctions for Γ supported in a tube along certain trajectories of the

flow. This localization method can be traced as far as to [31]. The idea of our construc-

tion goes back to the finite-dimensional case [7, 8, 9], but uses essentially different infinite

dimensional methods of [20, 37]. The choice of the trajectories is related to the existence

of Mane sequences θm ⊂ Θ which we define in this paper as an infinite dimensional

generalization of the Mane points mentioned above. It is easy to see that σap(E)∩T 6= ∅implies the existence of a Mane sequence. As a generalization of the Mane Lemma, we

show that the existence of a Mane sequence implies σap(E) ∩ T 6= ∅ under an additional

technical condition related to the residual spectrum.

Finally, we characterize in Section 5 the dichotomy of Φt by the existence and unique-

ness of bounded, continuous solutions to the mild inhomogeneous equation

u(ϕtθ) = Φt(θ)u(θ) +

∫ t

0

Φt−τ (ϕτθ)g(ϕtθ)dθ, t ≥ 0, θ ∈ Θ,

in C0(Θ;X) or Cb(Θ;X). In C0(Θ;X), this equation is just a reformulation of Γu = −g.

Our result gives an affirmative answer to a question remained unsolved in [23] and can

be regarded as a far-reaching generalization of a classical theorem by O. Perron, see

[13, 14, 23, 27, 32] for further comments and related results.

Acknowledgment. We thank A. Stepin for the suggestion to use ε-pseudo orbits which

was crucial in the proof of the perturbation theorem in Subsection 3.3, see [26].

Notation. We denote: T – unit circle in C; D – open unit disc in C; X – Banach space

with norm | · | ; L(X) – bounded linear operators on X; Ls(X) – bounded linear operators

on X with the strong topology; I – the identity operator; ρ(A), σ(A), σap(A), σr(A) –

resolvent set, spectrum, approximate point spectrum, residual spectrum of A.

2. The Dichotomy Theorem

We briefly introduce exponential dichotomy of cocycles (see the classical sources [17, 18,

42, 43] and [10, 44]) and evolution semigroups (see [21, 25, 40]), and prove the equivalence

of the exponential dichotomy of a cocycle and the hyperbolicity of the corresponding

evolution semigroup.

2.1. Exponential dichotomy. Let ϕtt∈R be a continuous flow on a locally compact

metric space Θ; that is, the map R × Θ 3 (t, θ) 7→ ϕtθ ∈ Θ is continuous and ϕt+τθ =

ϕt(ϕτθ), ϕ0θ = θ for all t, τ ∈ R and θ ∈ Θ. Let Φtt≥0 be an exponentially bounded,

strongly continuous cocycle over ϕtt∈R. This means that the map R+ × Θ 3 (t, θ) 7→Φt(θ) ∈ L(X) is strongly continuous, ‖Φt(θ)‖ ≤ Neωt for t ≥ 0, θ ∈ Θ, and some constants

N and ω, and the following cocycle property holds:

Φt+τ (θ) = Φt(ϕτθ)Φτ (θ) and Φ0(θ) = I for t, τ ≥ 0 and θ ∈ Θ. (2.1)4

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We will also consider cocycles for discrete time t ∈ N which are defined analogously.

Cocycles Φtt≥0 are in one-to-one correspondence with continuous linear skew-product

flows (LSPF) ϕtt≥0 defined by

ϕt : Θ×X → Θ×X : (θ, x) 7→ (ϕtθ,Φt(θ)x) for t ≥ 0.

Therefore, all assertions below concerning cocycles can be reformulated in terms of the

corresponding linear skew-product flows.

Definition 2.1. The cocycle Φtt≥0 has exponential dichotomy on Θ if there exists a

bounded, strongly continuous, projection valued function P : Θ→ Ls(X) satisfying

(a) P (ϕtθ)Φt(θ) = Φt(θ)P (θ),

(b) ΦtQ(θ) is invertible as an operator from ImQ(θ) to ImQ(ϕtθ) (with inverse

Φ−tQ (ϕtθ)),

(c) there exist constants β > 0 and M = M(β) ≥ 0 such that

‖ΦtP (θ)‖ ≤Me−βt and ‖[Φt

Q(θ)]−1‖ ≤Me−βt

for all θ ∈ Θ and t ≥ 0. The function P (·) is called dichotomy projection and the

constants β,M are the dichotomy constants.

Here and below, ΦtP (θ) and Φt

Q(θ) denote the restrictions Φt(θ)P (θ) : ImP (θ) →ImP (ϕtθ) and Φt(θ)Q(θ) : ImQ(θ)→ ImQ(ϕtθ), respectively, and we set Q = I − P for

a projection P . Notice that Φt(θ)Q(θ) is strongly continuous on R×Θ and satisfies (2.1)

with I replaced by Q(θ). Further, we define dichotomy along single orbits.

Definition 2.2. The cocycle Φtt∈K+ has exponential dichotomy (over K = R or Z) at

a point θ0 ∈ Θ if there are uniformly bounded projections Pτ ∈ L(X) for τ ∈ K which

depend strongly continuous on τ if K = R and satisfy

(a) Pτ+tΦt(ϕτθ0) = Φt(ϕτθ0)Pτ ,

(b) ΦtQ(ϕτθ0) is invertible from ImQτ to ImQτ+t,

(c) there exist constants β = β(θ0) > 0 and M = M(β) ≥ 0 such that

‖ΦtP (ϕτθ0)‖ ≤Me−βt and ‖[Φt

Q(ϕτθ0)]−1‖ ≤Me−βt

for t, τ ∈ K and t ≥ 0.

Clearly, dichotomy on Θ implies dichotomy at θ0. If ImQ(θ) and ImQ(ϕtθ) are finite

dimensional, then (b) and the second estimate in (c) follow from

|Φt(θ)x| ≥ 1Meβt|x| for x ∈ ImQ(θ), t ≥ 0.

If each operator Φt(θ) is invertible, exponential dichotomy at θ0 is equivalent to the

existence of a projection P such that

‖Φt(θ0)P [Φτ (θ0)]−1‖ ≤Me−β(t−τ), t ≥ τ,

‖Φt(θ0)Q[Φτ (θ0)]−1‖ ≤Me−β(τ−t), t ≤ τ,

cf. [40]. Then one has Pτ = Φτ (θ)P0[Φτ (θ)]−1. This is the classical definition of expo-

nential dichotomy, see [13, 14, 42]. We also recall the analogous concept for a single

operator.

Definition 2.3. A operator E ∈ L(X) is called hyperbolic if σ(E) ∩ T = ∅.5

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Observe that the powers Enn∈N or a semigroup Ett≥0 have exponential dichotomy

if and only if E or Et0 for some t0 > 0 are hyperbolic, respectively. In this case, the

dichotomy projection is given by the Riesz projection

P =1

2πi

∫T(λI − E)−1 dλ.

We denote by EtP = EtP and Et

Q = EtQ the restriction to the stable and the unstable sub-

space, ImP and ImQ, respectively; and analogously for the generator of the semigroup,

cf. [33, A-III].

For an exponentially bounded, strongly continuous cocycle Φt and λ ∈ C, we define

the rescaled cocycle Φtλ(θ) = e−λtΦt(θ) and the rescaled linear skew-product flow

ϕtλ : (θ, x) 7→ (ϕtθ, e−λtΦt(θ)x) for θ ∈ Θ, x ∈ X, t ≥ 0.

Note that the cocycle Φtλ corresponds to the equation u(t) = [A(ϕtθ)− λ]u(t).

Definition 2.4. The dynamical (or Sacker-Sell) spectrum Σ is defined by

Σ = λ ∈ R| Φtλt≥0 does not have exponential dichotomy.

2.2. Evolution semigroups. By F = C0(Θ;X) we denote the space of continuous func-

tions f : Θ→ X vanishing at infinity endowed with the sup-norm. For a cocycle Φtt≥0,

we introduce on C0(Θ;X) the evolution semigroup Ett≥0 by setting

(Etf)(θ) = Φt(ϕ−tθ)f(ϕ−tθ) for θ ∈ Θ, t ≥ 0, f ∈ C0(Θ;X).

One checks the following simple fact, cf. [21, Thm. 2.1], [40, Prop. 3].

Proposition 2.5. Ett≥0 is a strongly continuous semigroup on C0(Θ;X) if and only if

Φtt≥0 is an exponentially bounded, strongly continuous cocycle.

The following result is due to R. Rau, [40, Lemma 7], see [2, 20, 21, 25, 36] for earlier

and different versions.

Theorem 2.6 (Spectral Projection Theorem). Let Φtt≥0 be an exponentially bounded,

strongly continuous cocycle over a continuous flow ϕtt∈R on Θ. If the induced evolution

semigroup Et is hyperbolic on C0(Θ;X), then there exists a bounded, strongly continu-

ous, projection valued function P : Θ → Ls(X) such that the Riesz projection P for Et

corresponding to σ(Et) ∩ D is given by the formula (Pf)(θ) = P (θ)f(θ) for θ ∈ Θ and

f ∈ C0(Θ;X).

The idea of the proof is to check, see Lemma 3.2 in [20], that P commutes with the

multiplication operators induced by continuous, bounded functions χ : Θ → R. This

allows to define P (θ0)x = (Pf)(θ0) for θ0 ∈ Θ and x ∈ X, where f ∈ C0(Θ;X) is an

arbitrary function such that f(θ0) = x.

We now come to the first main result; our proof is a modification of [40, Thm. 10,12].

Theorem 2.7 (Dichotomy Theorem). Let Φtt≥0 be an exponentially bounded, strongly

continuous cocycle over a continuous flow ϕtt≥0 on a locally compact metric space Θ,

and let Ett≥0 be the induced evolution semigroup on C0(Θ;X). Then the cocycle has

exponential dichotomy on Θ if and only if the evolution semigroup is hyperbolic. In this

case, the Riesz projection P corresponding to the spectral set σ(E1)∩D and the dichotomy6

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projection P : Θ → Ls(X) of Φt are related by (Pf)(θ) = P (θ)f(θ) for θ ∈ Θ and

f ∈ C0(Θ;X).

Proof. (1) Assume that Φt has exponential dichotomy with projections P (θ). Define

P = P (·) on C0(Θ;X). Then PEt = EtP by (a) in Definition 2.1. The inequality

‖EtP‖L(C0(Θ;X)) ≤ Me−βt follows from the first inequality in (c). Also, Et

Q on ImQ has

the inverse (Rtf)(θ) = [ΦtQ(θ)]−1f(ϕtθ). Now, (c) yields ‖[Et

Q]−1‖L(C0(Θ;X)) ≤ Me−βt for

t ≥ 0. This means that Et is hyperbolic.

(2) Assume that Et is hyperbolic with Riesz projection P . The Spectral Projection

Theorem 2.6 shows that (Pf)(θ) = P (θ)f(θ) for some bounded, strongly continuous,

projection valued function P : Θ→ Ls(X). Since P is the Riesz projection for E1, EtQ is

invertible on ImQ and there exist constants β > 0 and M > 0 such that

‖EtP‖L(C0(Θ;X)) ≤Me−βt, t ≥ 0, (2.2)

‖EtQQf‖∞ ≥ 1

Meβt‖Qf‖∞, t ≥ 0. (2.3)

Clearly, (a) in Definition 2.1 holds. Let θ ∈ Θ and x ∈ X. Take a function f ∈ C0(Θ;X)

such that f(θ) = x and ‖f‖∞ = |x|. Using (2.2), we obtain

|ΦtP (θ)x| ≤ ‖Φt

P (·)f(·)‖∞ = ‖EtPf‖∞ ≤Me−βt‖f‖∞ = Me−βt|x|

which gives the first inequality in (c). To prove (b), fix t ≥ 0, θ ∈ Θ, and x ∈ Im Q(θ).

Recall that for any f ∈ Im Q we have ‖EtQf‖∞ = supθ∈Θ |Φt

Q(θ)f(θ)|. For any ε > 0

choose f ∈ Im Q such that f(θ) = x and ‖EtQf‖∞ ≤ |Φt(θ)x| + ε. This function can be

chosen as f = χx where χ is a continuous bump-function with χ(θ) = 1 and a sufficiently

small support. Then (2.3) implies

1Meβt|x| ≤ 1

Meβt‖f‖∞ ≤ ‖Et

Qf‖∞ ≤ |Φt(θ)x|+ ε,

and hence

|ΦtQ(θ)x| ≥ 1

Meβt|x|. (2.4)

In particular, the operator ΦtQ(θ) : ImQ(θ) → ImQ(ϕtθ) is injective. To see that it is

surjective, take y ∈ ImQ(ϕtθ). For a continuous, compactly supported function α : Θ 7→[0, 1] with α(ϕtθ) = 1, set f = α(·)Q(·)y. Then f ∈ ImQ. Since Et

Q is invertible on ImQ,

there exists g ∈ ImQ such that EtQg = f For x := g(θ) ∈ ImQ(θ0), this gives

ΦtQ(θ)x = Φt

Q(θ)g(θ) = f(ϕtθ) = α(ϕtθ)Q(ϕtθ)y = y.

Thus, (b) in Definition 2.1 holds. Together with (2.4) this establishes the second inequality

in (c) of this definition.

Corollary 2.8. The dichotomy projection is uniquely determined by Definition 2.1(a)–

(c).

Theorem 2.7 combined with the argument used in part (1) of its proof shows that the

existence of a dichotomy over R and Z for the cocycle Φt are equivalent:

Corollary 2.9. Given a cocycle Φtt≥0 the following assertions are equivalent:

(1) Φtt≥0 has an exponential dichotomy over R;

(2) the discrete time cocycle Φtt∈N has dichotomy.7

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The Dichotomy Theorem 2.7 relates our approach with the Sacker-Sell spectral theory,

[43], for infinite dimensional cocycles or linear skew-product flows. In fact, by rescaling

one obtains

Σ = log |σ(E1) \ 0|,cf. [25, 40]. One can also derive robustness of dichotomy from Theorem 2.7, see also

[10, 12, 13, 14, 18, 21, 22, 30, 45]. We present the following result in this direction.

Corollary 2.10. Assume that the exponentially bounded, strongly continuous cocycle

Φtt≥0 has exponential dichotomy. There exists ε > 0 such that every exponentially

bounded, strongly continuous cocycle Ψtt≥0 with

supθ∈Θ‖Φt(θ)−Ψt(θ)‖L(X) < ε

for some t > 0 also has exponential dichotomy.

Proof. Let Et and F t denote the evolution semigroups induced by Φt and Ψt,respectively. Assume that

supθ∈Θ‖Φt(θ)−Ψt(θ)‖L(X) < inf

λ∈T‖(λ− Et)−1‖−1 =: ε.

Then, using λ− F t = [1− (F t − Et)(λ− Et)−1] (λ− Et) and

‖Et − F t‖L(C0(Θ;X)) = supθ∈Θ‖Φt(θ)−Ψt(θ)‖L(X),

we derive that F t, and hence Ψt, has exponential dichotomy.

3. Global and Pointwise Dichotomies

We now want to characterize the (global) dichotomy of Φt on Θ by the (local) di-

chotomy at each θ0 ∈ Θ with uniform dichotomy constants. To that purpose, we develop

Banach algebra techniques which allow one to describe the dichotomy of Φt (or, equiv-

alently, the hyperbolicity of Et) in terms of the hyperbolicity of the discrete operators

πθ(E) on c0(Z;X) defined in (1.4).

3.1. Weighted translation algebras. For a continuous flow ϕtt∈R on a locally com-

pact metric space Θ, let us define the translation group (V tf)(θ) = f(ϕ−tθ) for t ∈ Ron the space C0(Θ;X). Set at(θ) = Φt(ϕ−tθ) for t ≥ 0, θ ∈ Θ, and an exponentially

bounded, strongly continuous cocycle Φtt≥0 over ϕtt∈R. We abbreviate V = V 1,

E = E1, ϕ = ϕ1, Φ = Φ1.

We denote by A = Cb(Θ;Ls(X)) the algebra of bounded, strongly continuous, operator

valued functions a = a(·) on Θ with the sup-norm ‖a‖A = ‖a‖∞ = supθ∈Θ ‖a(θ)‖L(X) and

point-wise multiplication. Clearly, A can be isometrically identified with the subalgebra A

of L(C0(Θ;X)) which contains the multiplication operators (denoted simply by a) given

by (af)(θ) = a(θ)f(θ) for θ ∈ Θ and f ∈ C0(Θ;X).

Let B denote the set of all sequences (ak)k∈Z with entries ak ∈ A satisfying (‖ak‖A)k∈Z ∈`1(Z). Equipped with the norm

‖(ak)k∈Z‖1 :=∞∑

k=−∞

‖ak‖A =∞∑

k=−∞

supθ∈Θ‖ak(θ)‖L(X) ,

8

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B is a Banach space. For b′ = (a′n)n∈Z and b′′ = (a′′n)n∈Z in B, we define b := b′ ∗ b′′ by

b = (an)n∈Z and an(θ) =∑

k∈Z a′k(θ)a

′′n−k(ϕ

−kθ) for θ ∈ Θ and n ∈ Z. Notice that

‖an(θ)‖L(X) ≤ ‖b′‖1 ‖b′′‖1.

The function θ 7→ a′k(θ)a′′n−k(ϕ

−kθ)x is continuous for x ∈ X and k, n ∈ Z. We have

|[an(θ1)− an(θ2)]x| ≤∑|k|≤N

| [a′k(θ1)a′′n−k(ϕ−kθ1)− a′k(θ2)a′′n−k(ϕ

−kθ2)]x |

+ 2 |x| ‖b′′‖1

∑|k|>N

‖a′k‖∞ .

for θ1, θ2 ∈ Θ and N ∈ N. Thus an ∈ A. We further estimate

‖b‖1 ≤∑

n

∑k‖a′k‖∞ ‖a′′n−k ϕ−k‖∞ =

∑k

∑n‖a′k‖∞ ‖a′′n−k‖∞ ≤ ‖b′‖1 ‖b′′‖1.

Therefore, (B, ∗, ‖ · ‖1) is a Banach algebra with the unit element e = e⊗ δn0. Here and

below, e(θ) = IX for θ ∈ Θ and δnk is the Kronecker delta. In other words, e = (en) where

e0 = I and en = 0 for n 6= 0.

To relate the algebra B with L(C0(Θ;X))), we introduce the (algebra) homomorphism

ρ : B→ L(C0(Θ;X))) : b = (ak)k∈Z 7→∞∑

k=−∞

akVk. (3.1)

Clearly, ρ(e) = I and ‖ρ(b)‖L(C0(Θ;X)) ≤ ‖b‖1 . We stress that, in general, ρ is not injective.

As an example, let ϕ1θ = θ for all θ ∈ Θ. Then, V = I and each a ∈ A ⊂ L(C0(θ;X))

can be represented as

a = 12a+ 1

2aV = 1

3a+ 2

3aV etc.

However, ρ is injective provided that ϕ is aperiodic, i.e., the set of aperiodic points for ϕ

is dense in Θ. Assuming, for a moment, the aperiodicity of ϕ, let B be the subspace of

L(C0(Θ;X)) containing the operators of the form

b =∞∑

k=−∞

akVk where ak ∈ A and ‖b‖1 :=

∞∑k=−∞

‖ak‖A <∞. (3.2)

For aperiodic ϕ, it was proved in Prop. 2.14 and 2.15 of [21] that the representation

b ∈ B as in (3.2) is unique and that (B, ‖ · ‖1) is a Banach algebra (with respect to the

composition of operators). Thus, ρ : B→ B is an isomorphism if ϕ is aperiodic.

Going back to a not necessarily aperiodic flow ϕ, let D ⊂ L(c0(Z;X)) be the algebra

of all bounded diagonal operators d = diag(d(n))n∈Z. Define

C := D =∞∑

k=−∞

dkSk ∈ L(c0(Z;X)) : dk ∈ D, ‖D‖C :=

∞∑k=−∞

‖dk‖L(c0(Z;X)) <∞,

where S : (xn)n∈Z 7→ (xn−1)n∈Z is the shift operator on c0(Z;X). The representation of

D ∈ C as D =∑dkS

k is unique since, for m0, n0 ∈ Z and x ∈ X with |x| = 1, one has

‖D‖L(c0(Z;X)) ≥ ‖D(x⊗ δn,n0)‖c0(Z;X) =∥∥∥∑

kdiag(d

(n)k )n∈Z (x⊗ δn−k,n0)n∈Z

∥∥∥c0(Z;X)

≥ |∑

kd

(m0)k x δm0−k,n0| = |d

(m0)m0−n0

x|.9

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We further need, for each θ ∈ Θ, the map

πθ :B→ L(c0(Z;X)) : b = (ak)n∈Z 7→∞∑

k=−∞

diag[ak(ϕnθ)]n∈Z S

k. (3.3)

Observe that πθ : B→ C and

‖πθ(b)‖C =∞∑

k=−∞

‖ diag(ak(ϕnθ))n∈Z‖L(c0(Z;X))

=∞∑

k=−∞

supn∈Z

supθ∈Θ‖ak(ϕnθ)‖L(X) =

∞∑k=−∞

‖ak‖∞ = ‖b‖1 .

Also, we compute for b = b′ ∗ b′′

πθ(b) =∞∑

k=−∞

diag

(∞∑

l=−∞

a′l(ϕnθ)a′′k−l(ϕ

n−lθ)

)n∈Z

Sk

=∞∑

l=−∞

[diag(a′l(ϕ

nθ))n∈Z SlS−l

∞∑k=−∞

diag(a′′k−l(ϕn−lθ))n∈Z S

k

]

=∞∑

l=−∞

[diag(a′l(ϕ

nθ))n∈Z Sl

∞∑k=−∞

diag(a′′k−l(ϕnθ))n∈Z S

k−l

]= πθ(b

′) · πθ(b′′).

As a result, πθ is a continuous homomorphism. Its importance relies on the following fact.

Proposition 3.1.⋂θ∈Θ ker πθ = 0.

Proof. Assume that πθ(b) = 0 for all θ ∈ Θ and some b = (an)n∈Z ∈ B. This means that

0 = πθ(b) =∑

kdiag(ak(ϕ

nθ))n∈Z Sk ∈ C

for θ ∈ Θ. Since the representation D =∑

k diag(d(n)k )Sk for D ∈ C is unique, we conclude

that ak(ϕnθ) = 0 for each k, n ∈ Z and θ ∈ Θ. Thus, b = 0 in B.

For a ∈ A, we define the operator b = I − aV ∈ L(C0(Θ;X)) and the sequence

b = (ak)k∈Z ∈ B by a0 = e, a1 = −a, and ak = 0 otherwise. Then ρ(b) = I − aV = b by

(3.1). Also, given b = I−aV , we introduce for each θ ∈ Θ the operator πθ(b) ∈ L(c0(Z;X))

by the rule

πθ(b) = πθ(I − aV ) := I − diag(a(ϕnθ))n∈ZS. (3.4)

To relate these concepts to the evolution semigroup Et on C0(Θ;X) induced by the

cocycle Φt, we remark that the operator I −E1 is of the form b = I −E = I − aV with

a(θ) = Φ(ϕ−1θ) for θ ∈ Θ. We can now give a sufficient condition for the invertibility of

I − E in terms of the operators πθ(b).

Theorem 3.2. Let b = I − aV for a ∈ A. Assume that πθ(b) is invertible in L(c0(Z;X))

for each θ ∈ Θ and that there exists a constant B > 0 such that

‖[πθ(b)]−1‖L(c0(Z;X)) ≤ B for all θ ∈ Θ. (3.5)

Then b−1 exists in L(C0(Θ;X)).10

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Proof. Let b = (ak)k∈Z ∈ B be given by a0 = I, a1 = −a, and ak = 0 for k 6= 0, 1. The

definitions (3.3) and (3.4) yield

πθ(b) = πθ(b) for θ ∈ Θ. (3.6)

Also, ρ(b) = b by (3.1). Setting d0(θ) = I, d1(θ) = diag(a(ϕnθ))n∈Z , and dk(θ) = 0 for

k 6= 1, 2 and θ ∈ Θ, we see that

πθ(b) =∑

kdk(θ)S

k ∈ C.

Next, we show that the inverse of πθ(b) also belongs to C. The proof of this fact is close

to that of Lemma 1.6 in [22].

Proposition 3.3. Under the assumptions of Theorem 3.2, the operator [πθ(b)]−1 =

[πθ(b)]−1 belongs to C for each θ ∈ Θ; that is,

[πθ(b)]−1 =

∞∑k=−∞

Ck(θ)Sk where Ck(θ) = diag(C

(n)k (θ))n∈Z satisfies (3.7)

∞∑k=−∞

supθ∈Θ‖Ck(θ)‖L(c0(Z;X)) <∞. (3.8)

Proof of Proposition 3.3. Set Dθ = πθ(b) = I − diag(a(ϕnθ))n∈ZS. Assumption (3.5) and

identity (3.6) yield

supθ∈Θ‖D−1

θ ‖ ≤ B. (3.9)

For γ > 0, we define the operator

Dθ(γ) = I − diag(eγ(|n|−|n−1|)a(ϕnθ))n∈Z S

on c0(Z;X). Note that Dθ(γ) = J−1γ DθJγ for Jγ = diag(e−γ|n|)n∈Z. Clearly,

‖Dθ −Dθ(γ)‖L(c0(Z;X)) ≤ max|1− eγ|, |1− e−γ| ‖a‖A → 0

as γ → 0. Choose γ0 such that

q := Bmax|1− e±γ0| ‖a‖A < 1.

Then ‖Dθ − Dθ(γ)‖ · ‖D−1θ ‖ ≤ q for all θ ∈ Θ and 0 ≤ γ ≤ γ0. Since Dθ is invertible

in L(c0(Z;X)), we conclude that Dθ(γ) is invertible and ‖[Dθ(γ)]−1‖ ≤ R := B1−q for all

0 ≤ γ ≤ γ0 and θ ∈ Θ.

Consider D−1θ as an operator matrix D−1

θ = [Ckj(θ)]k,j∈Z, where Ckj(θ) ∈ L(X) are

defined for x ∈ X by

Ckj(θ)x := (D−1θ (xj))k for xj = (δjnx)n∈Z.

From Dθ(γ)−1 = J−1γ D−1

θ Jγ we derive Dθ(γ)−1 = [eγ(|k|−|j|)Ckj(θ)]k,j∈Z. Now,

R ≥ ‖Dθ(γ)−1‖L(c0(Z;X)) ≥ eγ(|k|−|j|)‖Ckj(θ)‖L(X)

for k, j ∈ Z yields

‖Ck,0(θ)‖L(X) ≤ Re−γ|k| for θ ∈ Θ and k ∈ Z.

Observe that, for every j, k,m ∈ Z, one has

Dϕmθ = S−mDθSm and Ckj(ϕ

mθ) = Ck+m,j+m(θ). (3.10)11

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In particular, from Ck+m,m(θ) = Ck,0(ϕmθ) it follows that

‖Ck+m,m(θ)‖L(X) ≤ Re−γ|k| for k,m ∈ Z and θ ∈ Θ. (3.11)

So we can write D−1θ as

D−1θ =

∑kCk(θ)S

k =∑

kdiag(C

(n)k (θ))n∈ZS

k

for C(n)k (θ) = Cn,n−k(θ), and we estimate∑

k∈Z

supθ∈Θ‖Ck(θ)‖L(c0(Z;X)) =

∑k∈Z

supθ∈Θ

supn∈Z‖Cn,n−k(θ)‖L(X) ≤

∑k∈Z

Re−γ|k| <∞.

To conclude the proof of Theorem 3.2 we need the following fact.

Claim 1. θ 7→ C(0)k (θ), see (3.7), is a continuous, bounded function from Θ to Ls(X)

with ∑k∈Z

supθ∈Θ‖C(0)

k (θ)‖L(X) <∞.

We finish the proof of the theorem and show Claim 1 later. First, notice that (3.10)

implies

C(n)k (θ) = Cn,n−k(θ) = C0,−k(ϕ

nθ) = C(0)k (ϕnθ) (3.12)

for k, n ∈ Z and θ ∈ Θ. The sequence d = (C(0)k )k∈Z belongs to B by Claim 1. Thus, we

can apply to d the homomorphism ρ : B→ L(C0(Θ;X)). Let d = ρ(d). We want to show

that db = bd = I.

Consider the sequence r = d ∗ b − e ∈ B. Fix θ ∈ Θ and apply πθ : B → C. Using

(3.3), (3.12), and Proposition 3.3, we compute

πθ(r) = πθ(d) · πθ(b)− I =[∑

kdiag(C

(0)k (ϕnθ))n∈Z S

k]· πθ(b)− I

=[∑

kdiag(C

(n)k (θ))n∈Z S

k]· πθ(b)− I

=[∑

kCk(θ)S

k]· πθ(b)− I = 0.

Since θ ∈ Θ was arbitrary, Proposition 3.1 shows that r = 0 in B. As a consequence,

0 = ρ(r) = ρ(d ∗ b− e) = ρ(d) · ρ(b)− I = d · b− I.

Similarly, one obtains 0 = bd− I, and the theorem is proved.

Proof of Claim 1. Let k ∈ Z, x ∈ X, and θ0 ∈ Θ. Define x = (xn) ∈ c0(Z;X) by xn = x

if n = −k and xn = 0 if n 6= −k. Then, for every θ ∈ Θ,∣∣∣[C(0)k (θ)− C(0)

k (θ0)]x∣∣∣ =

∣∣∣∣∣∞∑

l=−∞

[C

(0)l (θ)− C(0)

l (θ0)]x−l

∣∣∣∣∣≤ sup

n∈Z

∣∣∣∑l

[C

(n)l (θ)− C(n)

l (θ0)]xn−l

∣∣∣=∥∥∥(∑

ldiag

(C

(n)l (θ)− C(n)

l (θ0))n∈Z

Sl)x∥∥∥c0(Z;X)

=∥∥∥([πθ(b)]

−1 − [πθ0(b)]−1)x∥∥∥c0(Z;X)

≤∥∥∥[πθ(b)]

−1∥∥∥L(c0(Z;X))

·∥∥∥[πθ0(b)− πθ(b)]y

∥∥∥c0(Z;X)

, (3.13)

12

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where y := [πθ0(b)]−1x ∈ c0(Z;X). It is easy to see that

Θ 3 θ 7→ πθ(b) ∈ Ls(c0(Z;X))

is (strongly!) continuous. So (3.13) and (3.9) imply the strong continuity of θ 7→ C(0)k (θ) ∈

Ls(X). ¿From (3.12) and (3.8) we derive∑k∈Z

supθ∈Θ‖C(0)

k (θ)‖L(X) =∑k∈Z

supθ∈Θ

supn∈Z‖C(0)

k (ϕnθ)‖L(X) =∑k∈Z

supθ∈Θ‖Ck(θ)‖L(c0(Z;X)) <∞,

and Claim 1 is proved.

Remark 3.4. Assume that ϕ is aperiodic. Then, ρ : B → B is an isomorphism and

πθ = πθ ρ−1. The proof of Theorem 3.2 shows that in this case b−1 belongs to B, cf. [21,

Lemma 4.4]. ♦

In the course of the proof of Theorem 3.2 we showed that for b = I−aV the element b is

invertible in B under condition (3.5). Its inverse is given by d = (C(0)k )k∈Z. Later, dealing

with eventually norm-continuous in θ cocycles, we will need the following refinement of

this fact. Let Anorm be the closed subalgebra Cb(Θ;L(X)) of A of all norm continuous,

bounded, operator valued functions a : Θ → L(X). Let Bnorm denote the corresponding

closed subalgebra of B, that is, Bnorm = (ak)k∈Z ∈ B : ak ∈ Anorm.

Corollary 3.5. Assume that a ∈ Cb(Θ;L(X)) is norm continuous. If the conditions of

Theorem 3.2 hold, then d = b−1 belongs to Bnorm.

Proof. We have to check that θ 7→ C(0)k (θ) ∈ L(X) is norm continuous (cf. Claim 1 in

the proof of Theorem 3.2). Fix k ∈ Z. Recall the estimate (3.13)

|[C(0)k (θ)− C(0)

k (θ0)]x| ≤ B ‖[πθ0(b)− πθ(b)]y‖c0(Z;X),

for x ∈ X and the constant B given in (3.5), and

y = [πθ0(b)]−1x =

∞∑l=−∞

diag(C(n)l (θ0))n∈Z S

lx = (C(n)n−k(θ0)x)n∈Z = (Cn,k(θ0)x)n∈Z,

where x = δn,−k ⊗ x and we have used (3.12). The inequality (3.11) yields ‖Cn,k(θ0)‖ ≤Re−γ|n−k| for n ∈ Z. Fix ε > 0 and take N = Nε such that Re−γ|n−k| < ε for |n| ≥ N .

Using the continuity of a : Θ→ L(X), choose δ > 0 such that

‖a(ϕnθ)− a(ϕnθ0)‖L(X) < ε provided |n| < N and d(θ, θ0) < δ.

For d(θ, θ0) < δ, we now conclude that

‖[πθ0(b)− πθ(b)]y‖c0(Z;X)

≤ max sup|n|<N

|[a(ϕnθ)− a(ϕnθ0)]Cn,k(θ0)x|, sup|n|≥N

|[a(ϕnθ)− a(ϕnθ0)]Cn,k(θ0)x|

≤ εmaxR, 2 supθ∈Θ‖a(θ)‖L(X) |x|.

13

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3.2. Discrete Dichotomy Theorem. The algebraic method developed above now en-

ables us to prove the main result of this section. This result connects global dichotomy

on Θ as defined in Definition 2.1, dichotomy at points θ ∈ Θ as defined in Definition 2.2,

the hyperbolicity of the operator given by (Ef)(θ) = Φ(ϕ−1θ)f(ϕ−1θ) on C0(Θ;X), and

the hyperbolicity of the operators πθ(E) = diag(Φ(ϕn−1θ))n∈Z S on c0(Z;X).

We start with some elementary facts. First, observe that

πθ(E)k = πθ(Ek) = diag(Φk(ϕn−kθ))n∈Z S

k (3.14)

for k ∈ N and θ ∈ Θ. We set ‖T‖•,Y = inf‖Ty‖ : ‖y‖ = 1 for a bounded operator T

on a Banach space Y . Observe that the operator πθ(E) is also defined on `∞(Z, X), the

space of bounded sequences endowed with the sup-norm.

Proposition 3.6. Let z ∈ T and θ ∈ Θ. We have ‖I − πθ(E)‖• = ‖zI − πθ(E)‖•on c0(Z;X) or `∞(Z, X). Further, the spectrum σ(πθ(E)) in c0(Z;X) or `∞(Z, X) is

invariant with respect to rotations centered at the origin, and if σ(πθ(E)) ∩ T = ∅, then

‖(I − πθ(E))−1‖ = ‖(zI − πθ(E))−1‖.

Proof. Define Lξ = diag(e−inξ)n∈Z on c0(Z;X) or `∞(Z, X) for ξ ∈ R. Then Lξ is an

invertible isometry on both spaces. So the identity

Lξ [zI − πθ(E)]L−1ξ = zI − e−iξ diag(Φ(ϕn−1θ))n∈ZS = e−iξ(zeiξI − πθ(E))

implies the result.

Recall that due to Corollary 2.9 the dichotomies of a cocycle for continuous and discrete

times are equivalent. Thus, we can work with discrete time.

Theorem 3.7 (Discrete Dichotomy Theorem). Let ϕ : Θ→ Θ be a homeomorphism on a

locally compact metric space Θ and let Φtt∈N with Φ1 = Φ ∈ Cb(Θ;Ls(X)) be a discrete

time cocycle over ϕ. The following assertions are equivalent.

(a) Φt has exponential dichotomy on Θ over Z.

(b) Φt has exponential dichotomy over Z at each point θ ∈ Θ with dichotomy con-

stants β(θ) ≥ β > 0 and M(β(θ)) ≤M .

(c) πθ(E) is hyperbolic on c0(Z;X) for each θ ∈ Θ and

supz∈T

supθ∈Θ‖[zI − πθ(E)]−1‖L(c0(Z;X)) <∞.

(d) E is hyperbolic on C0(Θ;X).

If (a)-(d) hold, let P (·) be the dichotomy projection for Φt on Θ, let Pn, n ∈ Z, be the

family of the dichotomy projections for Φt at θ ∈ Θ, let pθ ∈ L(c0(Z;X)), θ ∈ Θ, be the

Riesz projection for πθ(E) on c0(Z;X) that corresponds to σ(πθ(E))∩D, and let P be the

Riesz projection for E on C0(Θ;X) that corresponds to σ(E) ∩ D. Then,

P = P (·), Pn = P (ϕnθ), and pθ = diag(Pn)n∈Z.

Proof. (a) ⇒ (b). Let Φtt≥0 have exponential dichotomy for discrete time with projec-

tions P (·) ∈ Cb(Θ;Ls(X)) and constants β,M > 0, see Definition 2.1. For θ ∈ Θ, the

projections Pn := P (ϕnθ), n ∈ Z, clearly satisfy Definition 2.2 with constants β(θ) = β

and M(β(θ)) = M .14

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(b) ⇒ (c) Fix θ ∈ Θ. Let Pn, n ∈ Z, be the family of dichotomy projections for Φtat θ as in Definition 2.2. Define pθ = diag(Pn)n∈Z on c0(Z;X). By (a) in Definition 2.2,

we have pθπθ(E) = πθ(E)pθ. The identity (3.14) and Definition 2.2(b) imply that

(πθ(E)pθ)k = diag(Φk(ϕn−kθ)Pn−k)n∈Z S

k and

(πθ(E)qθ)−k = diag([Φk

Q(ϕnθ)Qn]−1)n∈Z S−k = diag(Φ−kQ (ϕn+kθ)Qn+k)n∈Z S

−k,

for k ∈ N. Thus, the estimates in Definition 2.2(c) give

‖(πθ(E)pθ)k‖L(c0(Z;X)) ≤M e−βk and ‖(πθ(E)qθ)

−k‖L(c0(Z;X)) ≤M e−βk.

As a result, πθ(E) is hyperbolic and (z, θ) 7→ ‖[zI − πθ(E)]−1‖L(c0(Z;X)) is bounded.

(c)⇒ (d). For every z ∈ T, apply Theorem 3.2 to b = I−aV , where a(θ) = z−1Φ(ϕ−1θ)

and (V f)(θ) = f(ϕ−1θ) for θ ∈ Θ.

(d) ⇒ (a) and P = P (·) can be shown exactly as Theorem 2.7.

One can improve the equivalence (b)⇔(c) as follows.

Corollary 3.8. The cocycle Φtt∈N has exponential dichotomy at θ ∈ Θ over Z if and

only if πθ(E) is hyperbolic on c0(Z;X).

Proof. Necessity was proved, in fact, in Theorem 3.7 (b)⇒(c). So let πθ(E) be hyperbolic.

As in [20, Lemma 3.2], we see that pθχ = χpθ for every χ = (χn) ∈ `∞(Z). By arguments

similar to those used in the proof of [40, Lemma 7], one obtains that pθ0 = diag(Pn)n∈Z for

a family of uniformly bounded projections Pn ∈ L(X), n ∈ Z. From pθπθ(E) = πθ(E)pθwe derive (a) in Definition 2.2. Further, Φk(ϕn−kθ) : ImQn−k → ImQn is surjective due

to (3.14) and the surjectivity of πθ(E)k on Im qθ. Finally, the estimates in (c) follow as in

the proof of Theorem 2.7.

In addition, for the case K = R, we can characterize pointwise dichotomy of the cocycle

by means of evolution semigroups along trajectories, Πtθt≥0 on C0(R;X), which are

defined by

(Πtθh)(s) = Φt(ϕs−tθ)h(s− t), s ∈ R, t ≥ 0, h ∈ C0(R;X),

for each θ ∈ Θ, cf. Theorem 5.9 of [21]. Since Φt is an exponentially bounded, strongly

continuous cocycle, the operators Uθ(s, τ) := Φs−τ (ϕτθ), s ≥ τ , yield an exponentially

bounded, strongly continuous evolution family, see [21, p.110]. In particular, (Πtθh)(s) =

Uθ(s, s − t)h(s − t) and Πtθ is an evolution semigroup on C0(R;X), see [20, 37] for

the definitions. By the results in [20, 36, 39], the evolution family Uθ(s, τ)s≥τ has

exponential dichotomy on R if and only if Πtθt≥0 is hyperbolic if and only if Γθ is

invertible. But dichotomy of Uθ(s, τ)s≥τ means exactly the dichotomy of Φtt≥0 at θ

over R. The next result now follows from the Dichotomy Theorem 2.7 and the techniques

used in the proof of Theorem 5.9 of [21].

Theorem 3.9. The following assertions are equivalent for an exponentially bounded,

strongly continuous cocycle Φtt≥0 over a continuous flow ϕtt∈R on Θ.

(a) Φtt≥0 has exponential dichotomy on Θ.

(b) σ(Π1θ) ∩ T = ∅ for each θ ∈ Θ and

supz∈T

supθ∈Θ‖[z − Π1

θ]−1‖L(C0(R;X)) <∞.

15

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(c) σ(Γθ) ∩ iR = ∅ for each θ ∈ Θ and

supξ∈R

supθ∈Θ‖[iξ − Γθ]

−1‖L(C0(R;X)) <∞.

(d) Φtt≥0 has exponential dichotomy over R at each point θ ∈ Θ with dichotomy

constants β(θ) ≥ β > 0 and M(β(θ)) ≤M .

Moreover, the Riesz projection P for the operator E, the Riesz projection pθ for the

operator Π1θ, the dichotomy projection P (·) for the dichotomy of Φt on Θ, and the

dichotomy projections P θτ τ∈R for the dichotomy of Φt at a point θ ∈ Θ are related as

follows:

P = P (·), P θτ = P (ϕτθ), (pθh)(τ) = P θ

τ h(τ), h ∈ C0(R;X).

3.3. Perturbation theorem. In this subsection we give an infinite-dimensional general-

ization of the Sacker-Sell Perturbation Theorem (see Theorem 6 in [43]) which establishes

the semicontinuity of the dynamical spectrum as a function of Θ. To this end, we assume

that Θ is a compact metric space and that the cocycle Φt is eventually norm continuous

in θ. This means (cf. [33, p. 38]) that for some t0 > 0 the map Θ 3 θ 7→ Φt0(θ) ∈ L(X)

is continuous in operator norm. By rescaling the time, we can assume that t0 = 1. Also,

we work with discrete time t ∈ N. We start with a preliminary fact.

Proposition 3.10. Assume that Θ 3 θ 7→ Φ1(θ) ∈ L(X) is norm continuous. If

σ(E)∩T = ∅, then Φnn∈N has exponential dichotomy with a norm continuous dichotomy

projection P (·).

Proof. Assume that E is hyperbolic. By the Discrete Dichotomy Theorem 3.7 we know

that the Riesz projection pθ for πθ(E) = πθ(aV ) is given by pθ = diag(P (ϕnθ))n∈Z. Notice

that a = Φ1 ϕ−1 ∈ Anorm, see the notations before Corollary 3.5. Define a = (an)n∈Z ∈Bnorm by a1 = a and an = 0 otherwise. Let bz = e − z−1a for z ∈ T. Recall that

πθ(bz) = πθ(I − z−1aV ) = I − z−1πθ(E) by (3.6). The Discrete Dichotomy Theorem 3.7

yields

‖(I − z−1πθ(E))−1‖L(c0(Z;X)) ≤ C for all θ ∈ Θ and z ∈ T.Applying Corollary 3.5 to bz = I − z−1aV , we see that for each z ∈ T the element

ze− a ∈ Bnorm is invertible in Bnorm. Define the idempotent p in the algebra Bnorm by

p =1

2πi

∫T(ze− a)−1dz.

Since πθ : B→ C is a continuous homomorphism for all θ ∈ Θ, we obtain

πθ(p) =1

2πi

∫Tπθ([ze− a]−1) dz =

1

2πi

∫T[zI − πθ(aV )]−1 dz = pθ

for every θ ∈ Θ. On the order hand, let p = (Pk)k∈Z for Pk ∈ Anorm. Then we have

0 = πθ(p)− pθ = πθ(P0)− pθ +∑k 6=0

πθ(Pk)Sk

by the definition (3.3) of πθ and πθ(Pk) = diag[Pk(ϕnθ)]n. Since the representation of

πθ(p) − pθ ∈ C as a series in powers of S is unique, we conclude that πθ(P0) = pθ and

πθ(Pk) = 0 for k 6= 0 and each θ ∈ Θ. So Proposition 3.1 yields Pk = 0 for k 6= 0, and

hence P = P0 ∈ Anorm (use the sequence (δn0Pk)n ∈ B). 16

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Let K be the set of all closed, ϕ-invariant subsets of Θ. For fixed Θ0 ∈ K and the

metric d(·, ·) on Θ, define

distΘ0(Θ1) = supθ1∈Θ1

infθ0∈Θ0

d(θ0, θ1) for Θ1 ∈ K.

Further, denote by Σ(Θ1) = Σ(ϕn|Θ1 × X) the Sacker-Sell dynamical spectrum, see

Definition 2.4, for the cocycle Φnn∈N restricted on Θ1, that is, for Φnn∈N = Φn(θ) :

n ∈ N, θ ∈ Θ1, or for the linear skew-product flow ϕn restricted to Θ1 ×X. The next

result implies the upper semicontinuity of the function Θ1 7→ Σ(Θ1) on K.

Theorem 3.11 (Perturbation Theorem). Let Φtt∈Z+ with Φ1 = Φ ∈ Cb(Θ;L(X)) be

a discrete time cocycle over a homeomorphism ϕ on a compact metric space Θ. Assume

that Φtt∈N has exponential dichotomy on the set Θ0 ∈ K in the sense of Definition

2.1. There exists δ∗ > 0 such that the cocycle Φtt∈N has exponential dichotomy on each

Θ1 ∈ K with the property

distΘ0(Θ1) ≤ δ for some δ ∈ (0, δ∗). (3.15)

The proof of the theorem uses nothing but elementary calculations with Neumann series

and the Discrete Dichotomy Theorem. Let us sketch our argument. For ε > 0, a sequence

θ = θnn∈Z ⊂ Θ is called ε-pseudo-orbit for ϕ if d(ϕθn, θn+1) ≤ ε for all n ∈ Z. As an

analogue to the operators πθ(E), we define the operator Tθ : (xn)n∈Z 7→ (Φ(θn−1)xn−1)n∈Zon c0(Z;X) for every ε-pseudo-orbit θ = θnn∈Z ⊂ Θ0.

Assuming that Φt has exponential dichotomy on Θ0, we will show that for sufficiently

small ε the operators I − Tθ are invertible on c0(Z, X) with uniformly bounded norms

‖(I − Tθ)−1‖ for all ε-pseudo-orbits θ ⊂ Θ0. Fix θ ∈ Θ1 and choose an ε-pseudo-orbit

θ = θn ⊂ Θ0 such that d(θn, ϕnθ) is small for each n ∈ Z. We use the continuity of Φ(·)

to show that for this choice of θn the norm ‖πθ(E)−Tθ‖L(c0(Z;X)) is small for sufficiently

small ε > 0. Since I−Tθ is invertible for θ ⊂ Θ0, we conclude that I−πθ(E) is invertible

and ‖(I −πθ(E))−1‖ ≤ C for each θ ∈ Θ1. Hence, Φt has exponential dichotomy on Θ1

by the Discrete Dichotomy Theorem.

Proof. Assume that Φt has exponential dichotomy on Θ0 ∈ K with projection P (·) and

constants M , β. Set c = maxθ∈Θ0‖P (θ)‖, 1. Choose γ > 0 such that

γ ≤ 116c. (3.16)

Fix N ∈ N such that Me−βN ≤ γ. Set ψ = ϕN and Ψ(θ) = ΦN(θ). We derive for all

θ ∈ Θ0 that

Ψ(θ)P (θ) = P (ψθ)Ψ(θ), (3.17)

ΨQ(θ) : ImQ(θ)→ ImQ(ψθ) is invertible, (3.18)

‖ΨP (θ)‖ ≤ γ, (3.19)

‖Ψ−1Q (θ)‖ ≤ γ. (3.20)

17

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Recall that P (·) and Ψ(·) are norm-continuous and Θ0 and Θ are compact. Fix ε∗ > 0

such that for every ε ∈ (0, ε∗) the following inequalities hold:

supd(θ,η)≤ε, θ,η∈Θ ‖Ψ(θ)−Ψ(η)‖ ≤ 3/16, (3.21)

supd(θ,η)≤ε, θ,η∈Θ0‖P (θ)− P (η)‖ ≤ 1

2(1+2c), (3.22)

c supd(θ,η)≤ε, θ,η∈Θ0‖P (θ)− P (η)‖ ·maxθ∈Θ ‖Ψ(θ)‖ ≤ 3/8. (3.23)

For a sequence θ = θnn∈Z ⊂ Θ0, we define by

Tθ : (xn) 7→ (Ψ(θn−1)xn−1), Pθ = diag P (θn), Qθ = I − Pθ, pθ = diag P (ψθn−1)

operators on c0(Z;X). Then (3.17) implies

TθPθ = pθTθ and TθQθ = (I − pθ)Tθ. (3.24)

The following lemma shows that exponential dichotomy, (3.17)–(3.20), implies the in-

vertibility of I−Tθ for every ε-pseudo-orbit θ for ψ in Θ0, where ε is given by (3.21)–(3.23).

Lemma 3.12. Let θ = θn ⊂ Θ0 be an ε-pseudo-orbit for ψ, where ε satisfies (3.21)–

(3.23). Then I − Tθ is invertible on c0(Z, X) and

‖(I − Tθ)−1‖ ≤ 83· (3.25)

We prove the lemma later and finish the proof of Theorem 3.11 first. Fix ε satisfying

(3.21)–(3.23). Fix δ∗ ≤ ε/4 such that d(ψθ, ψη) ≤ ε/2 as soon as d(θ, η) < 2δ∗. Let δ < δ∗

and let Θ1 ∈ K satisfy (3.15). For θ ∈ Θ1, define the operator Tθ := πθ(EN) : (xn) 7→

(Ψ(ψn−1θ)xn−1).

For a given θ ∈ Θ1 and each n ∈ Z, we use (3.15) to find θn ∈ Θ0 such that d(θn, ψnθ) <

2δ. The sequence θ = θn ⊂ Θ0 is an ε-pseudo-orbit for ψ because of

d(θn, ψθn−1) ≤ d(θn, ψnθ) + d(ψ(ψn−1θ), ψ(θn−1)) ≤ 2δ + ε/2 ≤ ε.

Thus, Lemma 3.12 can be applied to I − Tθ yielding ‖(I − Tθ)−1‖ ≤ 8/3. Also,

‖Tθ − Tθ‖ = supn∈Z‖Ψ(ψnθ)−Ψ(θn)‖ ≤ sup

d(ϑ,η)≤ε‖Ψ(ϑ)−Ψ(η)‖ ≤ 3/16 (3.26)

by (3.21). Then the operator

I − Tθ = I − Tθ − (Tθ − Tθ) = (I − Tθ)[I − (I − Tθ)−1(Tθ − Tθ)

]is invertible by (3.25) and (3.26). Moreover, the usual estimate of the Neumann series

implies ‖(I − Tθ)−1‖ ≤ 16/3.

As a consequence of the Discrete Dichotomy Theorem, EN is hyperbolic on C0(Θ1;X).

Hence, E is hyperbolic on C0(Θ1;X), and Φn has exponential dichotomy on Θ1 by the

Discrete Dichotomy Theorem.

Proof of Lemma 3.12. In the decomposition c0(Z, X) = ImPθ ⊕ ImQθ, we represent Tθas Tθ = A+B by

A =

[PθTθPθ 0

0 QθTθQθ

]and B =

[0 PθTθQθ

QθTθPθ 0

]. (3.27)

We prove the lemma in four steps.18

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Step 1. We show that

max‖PθTθPθ‖, ‖ [QθTθQθ]−1 ‖ ≤ 1/8. (3.28)

First, (3.16) and (3.19) imply

‖PθTθPθ‖ = supn∈Z‖P (θn)Ψ(θn−1)P (θn−1)‖ ≤ cγ ≤ 1/16 .

Further, by (3.24) one has

QθTθQθ = Qθ(I − pθ) · (I − pθ)TθQθ. (3.29)

The operator (I − pθ)TθQθ : ImQθ → Im (I − pθ) is invertible by (3.18) and (3.20), and

‖ [(I − pθ)TθQθ]−1 ‖ = sup

n∈Z‖Ψ−1

Q (θn)‖ ≤ γ. (3.30)

To prove that Qθ(I − pθ) : Im (I − pθ)→ ImQθ is invertible, we consider the operator

R := I − (Pθ −Qθ)(Pθ − pθ) = Qθ(I − pθ) + Pθpθ

on c0(Z, X) and its decomposition

R : Im pθ ⊕ Im (I − pθ)→ ImPθ ⊕ ImQθ ;

R =

[PθQθ

]R [pθ, I − pθ] =

[Pθpθ 0

0 Qθ(I − pθ)

].

Since θ = θn is an ε-pseudo-orbit for ψ, the inequality (3.22) yields

‖(Pθ −Qθ)(Pθ − pθ)‖ ≤ (1 + 2c)‖Pθ − pθ‖ = (1 + 2c) supn∈Z‖P (θn)− P (ψθn−1)‖ ≤ 1/2.

Therefore, R is invertible and, hence, Qθ(I − pθ) is invertible with ‖[Qθ(I − pθ)]−1‖ ≤

‖R−1‖ ≤ 2. This estimate together with (3.29)–(3.30) and (3.16) gives ‖ [QθTθQθ]−1 ‖ ≤

2γ ≤ 1/8.

Step 2. We show that I − A is invertible, and

‖(I − A)−1‖ ≤ 4/3 . (3.31)

The inequality (3.28) implies that the operators

Pθ − PθTθPθ and Qθ −QθTθQθ = −QθTθQθ

(Qθ − (QθTθQθ)

−1)

are invertible on ImPθ and ImQθ, respectively. Then

(I − A)−1 = (Pθ − PθTθPθ)−1 ⊕ (Qθ −QθTθQθ)−1 =

∞∑k=0

(PθTθPθ)k ⊕−

∞∑k=1

(QθTθQθ)−k.

As a result, ‖(I − A)−1‖ ≤ 4/3.

Step 3. We estimate ‖B‖ = max‖PθTθQθ‖, ‖QθTθPθ‖. By (3.24),

PθTθQθ = Pθ(Pθ − pθ)Tθ and QθTθPθ = (pθ − Pθ)pθTθ.

We also have ‖Pθ‖ ≤ c, ‖pθ‖ ≤ c, ‖pθ − Pθ‖ ≤ supd(θ,η)≤ε ‖P (θ) − P (η)‖, ‖Tθ‖ ≤supθ∈Θ ‖Ψ(θ)‖. Thus, (3.23) yields

‖B‖ ≤ c supd(θ,η)≤ε

‖P (θ)− P (η)‖ · supθ∈Θ‖Ψ(θ)‖ ≤ 3/8 . (3.32)

19

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Step 4. We use (3.31) and (3.32) to finish the proof. Indeed, ‖(I − A)−1B‖ ≤ 12

and

‖(I − Tθ)−1‖ = ‖[(I − A)(I − (I − A)−1B]−1‖≤ ‖(I − A)−1‖ ‖[I − (I − A)−1B]−1‖ ≤ 8/3.

We conclude this section with a typical application of Perturbation Theorem 3.11.

This theorem proves the roughness of dichotomy with respect to the perturbations which

are small in the metric on C(R;L(X)) (with the topology of uniform convergence on

compacta) versus the norm on Cb(R;L(X)), compare [43, §5].

Consider a variational equation

u(t) = [A+B(ϕtθ)]u(t), (3.33)

where A is a generator of a strongly continuous semigroup on X and B is a bounded

function such that

Θ 3 θ 7→ B(ϕ(·)θ) ∈ C(R;L(X)) is continuous. (3.34)

This means that

supt∈K‖B(ϕtθ1)−B(ϕtθ2)‖L(X) → 0 as θ1 → θ2

for each compact set K ⊂ R. Assume that (3.33) has exponential dichotomy at a point

θ0 ∈ Θ. Perturbation Theorem 3.11 implies that there exists an ε > 0 such that (3.33)

has dichotomy for each θ ∈ Θ with the property that the orbits O(θ) and O(θ0) satisfy

distO(θ0)(O(θ)) < ε. Indeed, the cocycle Φt(θ) for (3.33) satisfies

Φt(θ)x = etAx+

t∫0

e(t−τ)AB(ϕτθ)Φτ (θ)x dτ, x ∈ X,

see [10, Thm. 5.1]. Using Grownwall inequality, boundedness of B, and condition (3.34),

one checks that Θ 3 θ 7→ Φ1(θ) ∈ L(X) is norm continuous.

4. Annular Hull and Spectral Mapping Theorems

We now want to characterize the exponential dichotomy of the cocycle Φtt≥0 by the

spectrum of the generator Γ of the corresponding evolution semigroup Ett≥0 on the

space F = C0(Θ;X). To that purpose, we prove the spectral mapping and annular hull

theorems for Ett≥0. Assuming σap(E) ∩ T 6= ∅, we give an explicit construction of

approximate eigenfunctions for Γ. These approximate eigenfunctions are supported in

flow boxes selected to contain Mane sequences for Φnn∈Z+ . Throughout, Φtt≥0 is an

exponentially bounded, strongly continuous cocycle over a continuous flow ϕtt∈R on a

locally compact metric space Θ.

Definition 4.1. A sequence θm∞m=1 ⊂ Θ is called Mane sequence if there exist constants

C, c > 0 and vectors xm ∈ X such that

|Φm(θm)xm| ≥ c and (4.1)

|Φk(θm)xm| ≤ C for k = 0, 1, . . . , 2m. (4.2)

20

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We first discuss this concept, see also Corollary 4.12 and 4.13. Assume, for a moment,

that Φnn∈Z is a cocycle with invertible values and that θ0 is a Mane point, i.e., there

exists a nonzero vector x0 such that C := sup|Φn(θ0)x0| : n ∈ Z <∞, see the Introduc-

tion. We claim that θm = ϕ−mθ0 form a Mane sequence with vectors xm = Φ−m(θ0)x0.

Indeed,

|Φm(θm)xm| = |Φm(ϕ−mθ0)Φ−m(θ0)x0| = |x0| =: c > 0 and

|Φk(θm)xm| = |Φk(ϕ−mθ0)Φ−m(θ0)x0| = |Φk−m(θ0)x0| ≤ C

for k ∈ Z. Thus, Definition 4.1 extends the notion of a Mane point to the non-invertible

case.

Going back to the situation of a cocycle with possibly noninvertible values, we remark

the following.

Proposition 4.2. If Φtt∈N has exponential dichotomy on Θ, then there are no Mane

sequences in Θ.

Proof. Suppose, in the contrary, that θm∞1 is a Mane sequence with the corresponding

vectors xm. Decompose xm = xsm + xum, where xsm ∈ ImP (θm) and xum ∈ ImQ(θm). Let

ym = Φm(θm)xm, ysm = Φm(θm)xsm, and yum = Φm(θm)xum. Using dichotomy and (4.2) with

k = 2m and k = m, we estimate

|yum| ≤Me−βm|ΦmQ(ϕmθm)yum| ≤Me−βm|Φ2m(θm)xm|+ |Φm

P (ϕmθm)ysm|≤Me−βmC +Me−βm(|ym|+ |yum|) ≤Me−βmC +Me−βm(C + |yum|).

Thus, for sufficiently large m, we have

|yum| ≤Me−βmC +MCe−βm

1−M2e−2βm→ 0 as m→∞.

Together with (4.1) and (4.2) with k = m, this implies |ysm| ∈ [c/2, 2C] for large m. Since

Φt has exponential dichotomy, we conclude

c

2≤ |ysm| = |Φm(θm)xsm| ≤Me−βm|xsm| and 2C ≥ |yum| = |Φm(θm)xum| ≥M−1eβm|xum|.

Hence, |xsm| → ∞ and |xum| → 0 as m→∞. This contradicts (4.2) with k = 0.

We now give sufficient conditions for the existence of a Mane sequence, cf. [8, 29], using

the following refinement of Theorem 3.7 (c)⇒(d).

Lemma 4.3. Assume there exists a constant c > 0 such that

‖I − πθ(E)‖•,`∞(Z;X) ≥ c for all θ ∈ Θ.

Then we have ‖zI − E‖•,F ≥ c for all z ∈ T.

Proof. Let f ∈ F and z ∈ T. Notice that for each θ ∈ Θ the sequence xθ := (f(ϕkθ))k∈Zis bounded. The assumption and Proposition 3.6 yield

‖zf − Ef‖F = supθ∈Θ|zf(θ)− Φ1(ϕ−1θ)f(ϕ−1θ)| = sup

θ∈Θsupk∈Z|zf(ϕkθ)− Φ1(ϕk−1θ)f(ϕk−1θ)|

= supθ∈Θ‖[zI − πθ(E)]xθ‖`∞(Z;X) ≥ c sup

θ∈Θ‖xθ‖`∞(Z;X) = c ‖f‖F .

21

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Proposition 4.4. A Mane sequence θm∞1 exists provided one of the following conditions

holds.

(a) infθ∈Θ ‖I − πθ(E)‖•,`∞ = 0.

(b) σap(E) ∩ T 6= ∅.

Proof. In view of Lemma 4.3, it suffices to consider (a). Since Φt is exponentially

bounded, N := supθ ‖πθ(E)‖ is finite. Fix m ∈ N and let ε := (4∑2m

k=0 Nk)−1. By the

assumption, there exists σm ∈ Θ and y(m) = (y(m)l )l∈Z ∈ `∞(Z;X) such that ‖y(m)‖∞ = 1

and ‖[I − πσm(E)]y(m)‖∞ < ε. Hence,

12< ‖[πσm(E)]ky(m)‖`∞(Z;X) < 2 for k = 0, 1, . . . 2m.

For a fixed k ∈ 0, 1, . . . , 2m, we have

‖[πσm(E)]ky(m)‖l∞(Z;X) = supl∈Z|Φk(ϕl−kσm)y

(m)l−k | = sup

l∈Z|Φk(ϕl−mσm)y

(m)l−m|.

Select l such that 12≤ |Φm(ϕl−mσm)y

(m)l−m|. Note that |Φk(ϕl−mσm)y

(m)l−m| ≤ 2 holds for

k = 0, 1, . . . 2m. Therefore, θm := ϕl−mσm is a Mane sequence with the vectors xm :=

y(m)l−m.

In the construction of the approximate eigenfunctions we will use some basic facts

concerning the period function and flow boxes, or tubes.

A point θ ∈ Θ is called periodic with respect to the flow ϕtt∈R if ϕT θ = θ for some

T > 0, and aperiodic if there is no such T . The prime period function p : Θ→ R∪∞ is

defined as p(θ) = infT > 0 : ϕT θ = θ for periodic points θ and p(θ) =∞ for aperiodic

θ. Further, set

B(Θ) = θ ∈ Θ : p is bounded in a neighborhood of θ and

BC(Θ) = θ ∈ B(Θ) : p is continuous at θ.

Also, we denote by O(θ) = ϕtθ : t ∈ R the orbit through θ ∈ Θ. The following result

is taken from [6, Thm. IV.2.11].

Lemma 4.5. Let ϕtt∈R be a continuous flow on a locally compact, metric space Θ with

prime period function p. Suppose that θ0 ∈ Θ and τ are such that 0 < τ < p(θ0)/4. Then

there exists an open set U and a set Σ such that θ0 ∈ Σ ⊂ U , the closure U is compact,

and for each θ ∈ U there exists a unique number t = t(θ) in (−τ, τ) such that the point

σ := ϕ−tθ belongs to Σ. Moreover, the map θ 7→ t(θ) is continuous on U .

The set U as in Lemma 4.5 is called flow-box of length τ with cross-section Σ at θ0 ∈ Θ.

Lemma 4.6. For a continuous flow ϕtt∈R on a locally compact metric space Θ with

prime period function p, we have

(a) p is lower semicontinuous, in particular, the set θ : p(θ) ≤ d is closed for each

d ≥ 0;

(b) the points of continuity of p are dense in Θ;

(c) for θ0 ∈ BC(Θ) with p(θ0) > 0, there exists a relatively compact, open set Uand a set Σ 3 θ0 such that O(θ0) ⊂ U ⊂ BC(Θ), for θ ∈ U there is unique

t = t(θ) ∈ [0, p(θ)) with ϕ−tθ ∈ Σ, and U 3 θ 7→ t(θ) is continuous.22

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Proof. Assertion (a) is shown in [3, p.314]. The second assertion follows from (a) and [15,

pp.87]. To show (c), we use Lemma 4.5 to find a relatively compact, open flow box U0 at

θ0 of length τ ∈ (0, p(θ0)/4) with cross section Σ0 such that θ 7→ t(θ) is continuous, the

function p takes values in (3p(θ0)/4, 5p(θ0)/4) on U0, and

O(θ0) ∩ Σ0 = θ0. (4.3)

Let Σ1 be a relatively open subset of Σ0 containing θ0 such that Σ1 ⊂ U0. We claim that

O(θ0) has an open neighborhood U ⊂ U0 satisfying

for all θ ∈ U the set O(θ) ∩ Σ1 contains exactly one element. (4.4)

Suppose that no such neighborhood exists. Then there were points θn converging to a

point ϕtθ0 for some 0 ≤ t < p(θ0) such that O(θn)∩Σ1 is empty or contains more than one

element. But, since ϕ−tθn → θ0, we obtain ϕ−tθn ∈ U0 and, hence, points σn ∈ O(θn)∩Σ0

for large n. Moreover, t(ϕ−tθn) → t(θ0) = 0 so that σn → θ0. In particular, σn ∈ Σ1 for

large n. Then there must exist σ′n = ϕsnσn ∈ Σ1 for some sn with

0 < τ ≤ sn ≤ p(σn)− τ < p(θ0)− τ

2

and n sufficiently large, where we have used the continuity of p at θ0. Taking a subse-

quence, we may assume that sn → s so that

σ′n → ϕsθ0 ∈ Σ1 ⊂ U0 .

Again by the continuity of t(·), we derive ϕsθ0 ∈ Σ0. Due to (4.3), this yields ϕsθ0 = θ0.

But this contradicts 0 < s < p(θ0), and so there exists a neighborhood U of O(θ0)

satisfying (4.4).

It remains to show the continuity of p(·) and t(·) on U . So let θn → θ in U . Suppose

that p(θn) does not converge to p(θ). But, p(θn) → p0 for a subsequence, and so θn =

ϕp(θn)θn → ϕp0θ. Hence, p0 = kp(θ) for k 6= 1. This is impossible since p(θ), p0 ∈[3p(θ0)/4, 5p(θ0)/4]. Further, ϕrθn → ϕrθ in U0 for some r and sufficiently large n. This

yields the continuity of θ 7→ t(θ) on U .

Recall that Γ denotes the infinitesimal generator of the evolution semigroup Ett≥0 on

F = C0(Θ;X) defined by formula (1.1). Our next goal is to construct the approximate

eigenfunctions for Γ on F for a given Mane sequence. We first treat the case that the

Mane sequence consists of aperiodic points.

Lemma 4.7. Assume θm∞1 is a Mane sequence such that θm /∈ B(Θ) or p(θm) →∞. Then, for each ξ ∈ R, there exist functions fn ∈ D(Γ) with ‖fn‖∞ = 1 such that

‖(iξ − Γ)fn‖∞ → 0 as n→∞.

Proof. Without loss of generality, we assume that p(θm) → ∞ as m → ∞. Denote

n = minm,[

15p(θm)

], where [ · ] is the integer part. Note that n > 0 for sufficiently

large m. Set σ0 = ϕm(θm) and x = Φm−n(θm)xm. By the assumption there are constants

C, c > 0 such that c ≤ |Φm(θm)xm| and |Φk(θm)xm| ≤ C for k = 0, 1, . . . , 2m. This implies

that

0 < c ≤ |Φn(ϕ−nσ0)x| and |Φk(ϕ−nσ0)x| ≤ C for k = 0, 1, . . . , 2n. (4.5)23

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Using Lemma 4.5, we find an open flow box U of length n at σ0 with cross-section Σ 3 σ0

such that

|Φt(ϕ−nσ)x− Φt(ϕ−nσ0)x| ≤ 1 for 0 ≤ t ≤ 2n and σ ∈ Σ. (4.6)

For fixed n, choose “bump”-functions β ∈ C(U) with 0 ≤ β ≤ 1, β(σ0) = 1, and compact

support and γ ∈ C1(R) with 0 ≤ γ ≤ 1, supp γ ⊂ (0, 2n), γ(n) = 1, and ‖γ′‖∞ ≤ 2/n.

For ξ ∈ R, define

fn(θ) = e−iξtβ(σ)γ(t)Φt(ϕ−nσ)x if θ = ϕt−nσ ∈ U with t ∈ (0, 2n), σ ∈ Σ, (4.7)

and fn(θ) = 0 if θ /∈ U . Clearly, fn ∈ F and ‖fn‖∞ ≥ |fn(σ0)| ≥ c > 0 by (4.5). For

h > 0 so small that γ = 0 on [2n− h, 2n], one has

(Ehfn)(θ) = e−iξ(t−h)β(σ)γ(t− h)Φt(ϕ−nσ)x

for θ = ϕt−nσ ∈ U and (Ehfn)(θ) = 0 for θ /∈ U . Thus, fn ∈ D(Γ) and

(Γfn)(θ)− iξfn(θ) = −γ′(t)β(σ)e−iξtΦt(ϕ−nσ)x (4.8)

for θ = ϕt−nσ with t ∈ (0, 2n) and σ ∈ Σ and (Γfn)(θ) = 0 for θ /∈ U . Let N =

sup‖Φτ (θ)‖ : τ ∈ [0, 1], θ ∈ Θ. Then (4.6) and (4.5) imply

‖Γfn − iξfn‖∞ ≤2N

nmax

k=0,...,2n|Φk(ϕ−nσ)x| ≤ 2N(C + 1)

n.

If the Mane sequence consists of periodic points, we will make use of the following fact.

Lemma 4.8. Assume there is a sequence θm ⊂ BC(Θ) with 0 < d0 ≤ p(θm) ≤ d1 and

xm ∈ X such that |xm| = 1 and |[z − Φp(θm)(θm)]xm| → 0 as m → ∞ for some z ∈ T.

Then there exist ξ ∈ R and fn ∈ D(Γ) such that ‖fn‖∞ = 1 and ‖(iξ − Γ)fn‖∞ → 0 as

n→∞.

Proof. We may assume that z = eiη = eiη

p(θn)p(θn) and p(θn)→ p0 ∈ [d0, d1]. Set ξ = η/p0.

By Proposition 4.6, there exist open tubular neighborhoods Un of θn with sections Σn 3 θnsuch that on Un the function p is continuous and takes values in [d0/2, 2d1] and

|xn − e−iξp(σ)Φp(σ)(σ)xn| → 0 for σ ∈ Σn as n→∞. (4.9)

Take a continuous “bump”-function β : Un → [0, 1] with β(θn) = 1 and compact support.

Choose a function α ∈ C1[0, 1] satisfying 0 ≤ α ≤ 1, α = 0 on [0, 1/3], α = 1 on [2/3, 1],

and ‖α′‖∞ ≤ 4. Define

fn(θ) = e−iξtβ(σ)Φt(σ)[α(t/p(σ))xn + (1− α(t/p(σ)))e−iξp(σ)Φp(σ)(σ)xn] (4.10)

for θ = ϕtσ ∈ Un, where σ ∈ Σn and 0 ≤ t < p(σ), and fn(θ) = 0 for θ /∈ Un. Clearly,

fn ∈ F with

‖fn‖∞ ≥ |fn(θn)| = |Φp(θn)(θn)xn| ≥ 1/2

for large n. For 0 < h < d0/6, we compute

(Ehfn)(θ) =

e−iξ(t−h)β(σ)Φt(σ)[α( t−h

p(σ))xn + (1− α( t−h

p(σ)))e−iξp(σ)Φp(σ)(σ)xn], t ≥ h,

e−iξ(t−h)β(σ)Φt(σ)e−iξp(σ)Φp(σ)(σ)xn, t < h,24

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for θ = ϕtσ ∈ Un and (Ehfn)(θ) = 0 otherwise. (Write ϕ−hθ = ϕp(θ)+t−hσ if θ ∈ Un and

t− h < 0.) Thus, fn ∈ D(Γ) and

Γfn(θ)− iξfn(θ) = −e−iξtβ(σ)1

p(σ)α′( t

p(σ))Φt(σ)[xn − e−iξp(σ)Φp(σ)(σ)xn] (4.11)

for θ = ϕtσ ∈ Un and zero otherwise. Let N = sup‖Φτ (θ)‖ : θ ∈ Θ, 0 ≤ τ ≤ 2d1.Finally we obtain

‖(Γ− iξ)fn‖∞ ≤8N

d0

|xn − e−iξp(σ)Φp(σ)(σ)xn| → 0

as n→∞ thanks to (4.9).

We are now in the position to prove the annular hull theorem and the spectral mapping

theorem for the evolution semigroup

(Etf)(θ) = Φt(ϕ−tθ)f(ϕ−tθ) on F = C0(Θ;X)

with generator Γ. The spectral mapping theorem says that

σ(Et)\0 = exp tσ(Γ) for t ≥ 0. (4.12)

In general, this result does not hold if ϕt is not aperiodic, see e.g. [9, Ex. 2.3]. The annular

hull theorem states that

exp tσ(Γ) ⊆ σ(Et)\0 ⊆ H(exp tσ(Γ)) for t ≥ 0,

where H(·) is the annular hull of the set (·), that is, the union of all circles centered at the

origin that intersect the set (·). In other words, the annular hull theorem can be written

as

T · (σ(Et)\0) = T · exp tσ(Γ) for t ≥ 0. (4.13)

Observe that the annular hull theorem can be violated if there are fixed points. Indeed,

for the one-point set Θ = θ and identical flow ϕt(θ) ≡ θ define a cocycle Φt(θ) = etA

for a strongly continuous semigroup etA on X which does not satisfy the annular hull

theorem (see [33, A-III] for examples). Clearly, in this case (4.13) does not hold for the

corresponding evolution semigroup.

Theorem 4.9 (Spectral Mapping/Annular Hull Theorem). Let Ett≥0 be the evolu-

tion semigroup on F = C0(Θ;X) with generator Γ induced by an exponentially bounded,

strongly continuous cocycle Φtt∈R+ over a continuous flow ϕtt∈R on a locally compact

metric space Θ, and let p be the function of prime periods for ϕt.(a) If B(Θ) = ∅, then σ(Γ) is invariant with respect to vertical translations, σ(Et)

is invariant with respect to rotations centered at zero, and the spectral mapping

theorem (4.12) holds.

(b) If p(θ) ≥ d0 > 0 for all θ ∈ B(Θ), then the annular hull theorem (4.13) holds.

Proof. By virtue of the spectral inclusion theorem and the spectral mapping theorem

for the residual spectrum, see e.g. [33, A-III.6], and a standard rescaling, (4.13) holds

provided that

σap(E) ∩ T 6= ∅ implies σ(Γ) ∩ iR 6= ∅25

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for E = E1. So let z ∈ σap(E) ∩ T. By Proposition 4.4, there exists a Mane sequence

θm in Θ. We have to consider two cases:

Aperiodic Case: there is a subsequence with θm /∈ B(Θ) or p(θm)→∞ as m→∞;

Periodic Case: θm ∈ B(Θ) and p(θm) < d1 for a constant d1 .

In the aperiodic case, Lemma 4.7 shows that iR ⊆ σap(Γ) which implies (4.12). The

other assertions in (a) then follow from the spectral inclusion theorem [33, A-III.6.2] and

∂σ(Γ) ⊂ σap(Γ).

In the periodic case, using Lemma 4.6, we may and will assume that θm ∈ Ω := θ ∈BC(Θ) : d0 ≤ p(θ) ≤ d1. The set Ω is a ϕt-invariant and locally compact metric space by

Lemma 4.6. On Ω, we define a new flow ψtt∈R by ψt(θ) = ϕtp(θ)(θ) and a new cocycle

Ψtt∈R+ over ψt by Ψt(θ) = Φtp(θ)(θ). Recall that θ 7→ p(θ) is continuous on BC(Θ).

Clearly, ψt is continuous and Ψt is strongly continuous and exponentially bounded

on Ω. So there exists the induced evolution semigroup EtΨ on C0(Ω;X). Let EΨ = E1

Ψ.

We stress that (EΨf)(θ) = Φp(θ)(θ)f(θ) for f ∈ C0(Ω;X) and θ ∈ Ω. The following fact

is stated as a separate lemma for later use.

Lemma 4.10. Let Ω be as defined above. Assume that σ(Φp(θ)(θ))∩T = ∅ for each θ ∈ Ω

and

supθ∈Ω‖[zI − Φp(θ)(θ)]−1‖L(X) <∞ for all z ∈ T.

Then Φt has exponential dichotomy on Ω.

Proof of Lemma 4.10. The assumptions imply σ(EΨ) ∩ T = ∅, where

(zI − EΨ)−1f(θ) = [zI − Φp(θ)(θ)]−1f(θ)

for θ ∈ Ω, z ∈ T, and f ∈ C0(Ω;X). An application of the Dichotomy Theorem 2.7 to

EΨ shows that the cocycle Ψt has exponential dichotomy on Ω with the dichotomy

projection P (·) and the dichotomy constants β,M > 0. Therefore, the cocycle Φt has

exponential dichotomy on Ω with P (·), β/d1, and M .

Since Ω contains a Mane sequence, the cocycle Φt does not have exponential di-

chotomy on Ω due to Proposition 4.2. Therefore the assumptions in Lemma 4.10 cannot

be satisfied. This means that either

(1) ‖zI − Φp(θn)(θn)‖•,X → 0 for some z ∈ T and θn ∈ Ω or

(2) there exist z ∈ T, θ0 ∈ Ω, y ∈ X, and δ > 0 such that |zx − Φp(θ0)(θ0)x − y| ≥ δ

for all x ∈ X.

In case (1), Lemma 4.8 implies σap(Γ)∩ iR 6= ∅. In case (2), take a function g ∈ C0(Θ;X)

such that g(θ0) = y. For all f ∈ C0(Θ;X), we have

‖zf − Ep(θ0)f − g‖∞ ≥ |zf(θ0)− Φp(θ0)f(θ0)− y| ≥ δ;

that is, z ∈ σr(Ep(θ0))∩T. Hence, σr(Γ)∩ iR 6= ∅ by the spectral mapping theorem for the

residual spectrum [33, A-III.6.3]. Altogether, we have established (b) in Theorem 4.9.

Combining the above result with the Dichotomy Theorem 2.7, we obtain the following.26

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Corollary 4.11. Let Φtt∈R+ be an exponentially bounded, strongly continuous cocycle

Φtt∈R+ over a continuous flow ϕtt∈R on a locally compact metric space Θ, let p(θ) ≥d0 > 0 for all θ ∈ B(Θ), and let Γ be the generator of the induced evolution semigroup on

C0(Θ;X). Then the cocycle has exponential dichotomy on Θ if and only if iR ⊂ ρ(Γ). If

B(Θ) = ∅, then these conditions are equivalent to ρ(Γ) ∩ iR 6= ∅.

We have seen in Proposition 4.4 that the condition σap(E)∩T 6= ∅ implies the existence

of a Mane sequence. We can now give two results in the opposite direction.

Corollary 4.12. Assume that σr(E) ∩ T = ∅. Then, σap(E) ∩ T 6= ∅ if and only if there

exists a Mane sequence θm∞1 .

Proof. If σap(E) ∩ T 6= ∅, then a Mane sequence exists by Proposition 4.4. If a Mane

sequence exists, then σ(E) ∩ T 6= ∅ by Proposition 4.2 and Dichotomy Theorem 2.7. So

the assumption, σr(E) ∩ T = ∅, implies the result.

Corollary 4.13. Assume that θm∞1 is a Mane sequence.

(a) If the aperiodic case holds for θm∞1 , that is, there is a subsequence such that

θm /∈ B(Θ) or p(θm)→∞ as m→∞, then T ⊂ σap(E).

(b) If the periodic case holds for θm∞1 , that is, θm ∈ Ω = θ ∈ BC(θ) : d0 ≤p(θ) ≤ d1 for some constant d1, and if σr(Φ

p(θ)(θ)) ∩ T = ∅ for all θ ∈ Ω, then

σap(E) ∩ T 6= ∅.

Proof. In the aperiodic case (a), one has iR ⊂ σap(Γ) for the generator Γ of the evolution

semigroup Et on C0(Θ;X) by Lemma 4.7. So the spectral inclusion theorem yields

T ⊂ σap(E).

In the periodic case (b), the cocycle Φt does not have exponential dichotomy on Ω by

Proposition 4.2. Therefore, the assumptions of Lemma 4.10 do not hold. Since we have

assumed σr(Φp(σ)(σ)) ∩ T = ∅ for all θ ∈ Ω, we conclude that ‖zI − Φp(θn)(θn)‖•,X → 0

for some z ∈ T and θn ∈ Ω. Lemma 4.8 implies iξ ∈ σap(Γ) for some ξ ∈ R, and hence

σap(E) ∩ T 6= ∅ by the spectral inclusion theorem.

5. Dichotomy and Mild Solutions

In the sequel, we relate dichotomy of the cocycle Φt or the linear skew-product flow

ϕt on Θ × X to the existence and uniqueness of the solution u for the mild form of

the inhomogeneous variational equation u(ϕtθ) = (A(ϕtθ) − λ)u(ϕtθ) + g(ϕtθ). Recall

that Φtλ(θ) = e−λtΦt(θ) for λ ∈ C, t ≥ 0, and θ ∈ Θ. Consider the mild inhomogeneous

equation

u(ϕtθ) = Φtλ(θ)u(θ) +

∫ t

0

Φt−τλ (ϕτθ)g(ϕτθ) dτ, t ≥ 0, θ ∈ Θ, (5.1)

on C0(Θ;X) or Cb(Θ;X), the space of bounded, continuous functions f : Θ→ X endowed

with the sup-norm.

Definition 5.1. Let F ∈ C0(Θ;X), Cb(Θ;X) and λ ∈ C. We say that condition

(Mλ,F) holds if for each g ∈ F there exists a unique u ∈ F satisfying (5.1).

If the condition (Mλ,F) holds for some λ ∈ C, then we can define a linear operator

Rλ : g 7→ u on F that recovers the unique solution u of (5.1) for a given g. This mapping

is, in fact, continuous.27

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Lemma 5.2. If condition (Mλ,F) holds for some λ ∈ C and F ∈ C0(Θ;X), Cb(Θ;X),then Rλ is bounded on F .

Proof. Thanks to the Closed Graph Theorem, it suffices to show the closedness of Rλ.

Take gn, g, u ∈ F such that gn → g and un := Rλgn → u in F . Since (5.1) holds for unand gn, we have that u = Rλg:

u(θ) = limn→∞

un(θ) = limn→∞

[Φtλ(ϕ

−tθ)un(ϕ−tθ) +

∫ t

0

Φt−τλ (ϕτ−tθ)gn(ϕτ−tθ) dτ

]= Φt

λ(ϕ−tθ)u(ϕ−tθ) +

∫ t

0

Φt−τλ (ϕτ−tθ)g(ϕτ−tθ) dτ.

First, we show by a standard argument that the exponential dichotomy of Φt implies

(Mλ,F). Here we do not need to assume that the cocycle is exponentially bounded.

Proposition 5.3. Let Φtt∈R+ be a strongly continuous cocycle Φtt∈R+ over a contin-

uous flow ϕtt∈R on a locally compact metric space Θ. Assume that Φt has exponential

dichotomy on Θ. Then condition (Mλ,F) holds for F = C0(Θ;X) or Cb(Θ;X) and for

all λ = iξ ∈ iR.

Proof. If Φt has exponential dichotomy, then Φtλ with λ = iξ ∈ iR also has it, so

it is enough to prove the theorem for λ = 0. Let P (·) be the dichotomy projection and

M,β > 0 be the dichotomy constants from Definition 2.1. Define Green’s function G(θ, t)

by

G(θ, t) =

ΦtP (θ) = Φt(θ)P (θ), t ≥ 0, θ ∈ Θ,

−ΦtQ(θ) = −[Φ−tQ (ϕtθ)Q(ϕtθ)]−1, t < 0, θ ∈ Θ.

Clearly, ‖G(θ, t)‖ ≤ Me−β|t| and (θ, t) 7→ G(θ, t) is strongly continuous on Θ × (R\0).Moreover, Green’s operator G defined by

(Gf)(θ) =

∫ ∞−∞

G(ϕ−τθ, τ)f(ϕ−τθ) dτ for f ∈ F and θ ∈ Θ

is a bounded operator on C0(Θ;X) and Cb(Θ;X). To see that G indeed maps C0(Θ;X)

into C0(Θ;X), fix f ∈ C0(Θ;X) having support in a compact subset K of Θ, and let

θn /∈ K tend to ∞. Set

tn := supt ≥ 0 : ϕτ (θn) /∈ K for all τ ∈ [−t, t].

Suppose that tn < T < ∞. Then there exists τn ∈ [−T, T ] such that ϕτnθn ∈ K for

n ∈ N. This means that θn ∈⋃|τ |≤T ϕ

τ (K), a contradiction. As a result, tn → ∞ and

(Gf)(θn)→ 0 due to the estimate ‖G(θ, τ)‖ ≤Me−β|τ |.28

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Let u = Gg for g ∈ F . We compute

u(ϕtθ)− Φt(θ)u(θ)

=

∫ ∞0

Φτ (ϕt−τθ)P (ϕt−τθ)g(ϕt−τθ) dτ − Φt(θ)

∫ ∞0

Φτ (ϕ−τθ)P (ϕ−τθ)g(ϕ−τθ) dτ

−∫ 0

−∞ΦτQ(ϕt−τθ)Q(ϕt−τθ)g(ϕt−τθ)d τ + Φt(θ)

∫ 0

−∞ΦτQ(ϕ−τθ)Q(ϕ−τθ)g(ϕ−τθ) dτ

=

∫ t

−∞Φt−τ (ϕτθ)P (ϕτθ)g(ϕτθ)dτ −

∫ 0

−∞Φt−τ (ϕτθ)P (ϕτθ)g(ϕτθ) dτ

−∫ ∞t

Φt−τQ (ϕτθ)Q(ϕτθ)g(ϕτθ) dτ +

∫ ∞0

Φt−τQ (ϕτθ)Q(ϕτθ)g(ϕτθ) dτ

=

∫ t

0

Φt−τ (ϕτθ)P (ϕτθ)g(ϕτθ) dτ +

∫ t

0

Φt−τ (ϕτθ)Q(ϕτθ)g(ϕτθ) dτ

=

∫ t

0

Φt−τ (ϕτθ)g(ϕτθ)dτ

for t ≥ 0 and θ ∈ Θ. This proves that u = Gg satisfies (5.1) with λ = 0.

To show uniqueness, take g = 0 and let u ∈ F satisfy u(ϕtθ) = Φt(θ)u(θ) for θ ∈ Θ

and t ≥ 0. Since Φt has exponential dichotomy, we have

P (θ)u(θ) = ΦtP (ϕ−tθ)P (ϕ−tθ)u(ϕ−tθ) and Q(θ)u(θ) = [Φt

Q(θ)]−1Q(ϕtθ)u(ϕtθ)

for t ≥ 0 and θ ∈ Θ. Thus, the estimates

|P (θ)u(θ)| ≤Me−βt‖u‖∞ and |Q(θ)u(θ)| ≤Me−βt‖u‖∞imply u = 0.

We now address the converse of Proposition 5.3. In the case F = C0(Θ;X), the result

is an easy consequence of Corollary 4.11.

Theorem 5.4. Let Φtt∈R+ be an exponentially bounded, strongly continuous cocycle

over a continuous flow ϕtt∈R on a locally compact metric space Θ.

(a) Assume that p(θ) ≥ d0 > 0 for all θ ∈ B(Θ). If condition (Mλ, C0(Θ;X)) holds

for all λ = iξ ∈ iR, then Φt has exponential dichotomy on Θ.

(b) Assume that B(Θ) = ∅. If condition (M0, C0(Θ;X)) holds, then Φt has expo-

nential dichotomy on Θ.

Proof. Due to Corollary 4.11, we only have to show that iR ⊂ ρ(Γ) in case (a) and

0 ∈ ρ(Γ) in case (b). Fix g ∈ C0(Θ;X). By the assumption, there exists the function

u = Rλg ∈ C0(Θ;X). Applying Eh in (5.1), we obtain

e−λh(Ehu)(θ) = Φhλ(ϕ

−hθ)u(ϕ−hθ) = u(θ)−∫ h

0

Φh−τλ (ϕτ−hθ)g(ϕτ−hθ) dτ.

Therefore,

1h[e−λh(Ehu)(θ)− u(θ)] = −1

h

∫ h

0

Φτλ(ϕ

−τθ)g(ϕ−τθ) dτ,

and we conclude that u ∈ D(Γ) and (λ − Γ)u = g. This means that λ − Γ is surjective

for all λ ∈ iR in case (a) and for λ = 0 in case (b).29

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If (Γ − λI)u = 0, then e−λtEtu = u or u(ϕtθ) = Φtλ(θ)u(θ) for θ ∈ Θ and t ≥ 0. So

the uniqueness part of the condition (Mλ, C0(Θ;X)) implies the injectivity of λ − Γ for

all λ ∈ iR in case (a) and for λ = 0 in case (b).

The case F = Cb(Θ;X) cannot be reduced to the results of the previous sections since

the evolution semigroup is not strongly continuous on Cb(Θ;X). Instead, we use the

discrete operators πθ(E). We start with two preliminary lemmas. The first one can be

proved exactly as [20, Lemma 2.4].

Lemma 5.5. Assume that θ ∈ Θ satisfies ϕ1θ = θ. Then, πθ(E) is hyperbolic on c0(Z;X)

if and only if Φ1(θ) is hyperbolic on X. Moreover, if 1 /∈ σap(πθ(E)), then σap(Φ1(θ))∩T =

∅.

Lemma 5.6. Assume that condition (Mλ,F) holds for F ∈ C0(Θ;X), Cb(Θ;X) and

some λ ∈ C. Then, for each periodic point θ0 ∈ Θ with p(θ0) > 0, the operator eλp(θ0) −Φp(θ0)(θ0) is surjective in X.

Proof. Fix y ∈ X. Take β ∈ C([0, p(θ0)];R) such that β(0) = 0, β(p(θ0)) = 1/p(θ0), and∫ p(θ0)

0β(τ)dτ = 1. On the orbit O(θ0) define

g(ϕτθ0) = β(τ)Φτλ(θ0)y +

[1

p(θ0)− β(τ)

]Φp(θ0)+τλ (θ0)y.

The function g is continuous on O(θ0). Extend g on Θ continuously to a function g ∈ Fwith a compact support. Let u = Rλg. Observe that

Φp(θ0)−τλ (ϕτθ0)Φ

p(θ0)+τλ (θ0) = Φ2p(θ0)(θ0).

Using this fact, (5.1), and∫ p(θ0)

0[ 1p(θ0)− β(τ)]dτ = 0, one obtains

u(ϕp(θ0)θ0) = u(θ0) = Φp(θ0)λ (θ0)u(θ0) + Φ

p(θ0)λ (θ0)y.

For x = u(θ0), we get

x− e−λp(θ0)Φp(θ0)(θ0)x = e−λp(θ0)Φp(θ0)(θ0)y.

Therefore, [eλp(θ0) − Φp(θ0)(θ0)](e−λp(θ0)(x+ y)) = y.

Theorem 5.7. Let Φtt∈R+ be an exponentially bounded, strongly continuous cocycle

over a continuous flow ϕtt∈R on a locally compact metric space Θ.

(a) Assume that p(θ) ≥ d0 > 0 for all θ ∈ B(θ). If condition (Mλ, Cb(Θ, X)) holds for

all λ = iξ ∈ iR, then Φt has exponential dichotomy on Θ.

(b) Assume that B(Θ) = ∅. If condition (M0, Cb(Θ;X)) holds, then Φt has exponen-

tial dichotomy on Θ.

Proof. We will show that I − πθ(E) is invertible in L(c0(Z;X)) and

‖(I − πθ(E))−1‖L(c0(Z;X)) ≤ C for θ ∈ Θ. (5.2)

Then the theorem follows from the Discrete Dichotomy Theorem 3.7 and Proposition 3.6.

Step I. We prove that

infθ∈Θ ‖I − πθ(E)‖•,c0(Z;X) > 0. (5.3)30

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Suppose (5.3) does not hold; that is, suppose that

infθ∈Θ ‖I − πθ(E)‖•,c0(Z;X) = 0. (5.4)

By Proposition 4.4, there exists a Mane sequence θm in Θ. We consider separately the

aperiodic and the periodic cases.

(1) Assume that there is a subsequence with p(θm)→∞ or θm /∈ B(Θ). An application

of Lemma 4.7 gives fn ∈ D(Γ) such that ‖fn‖∞ = 1 but Γfn → 0, where Γ is the generator

of the corresponding evolution semigroup on C0(Θ;X). Recall that the functions fn are

given explicitly by (4.7) while Γfn is computed in (4.8). Using (5.1), it is easy to verify

that R0Γfn = −fn. This contradicts the continuity of R0 shown in Lemma 5.2.

(2) It remains to consider the case θm ∈ Ω = θ ∈ BC(Θ) : d0 ≤ p(θ) ≤ d1 for some

d1 (use Proposition 4.6). We claim that there is a constant c such that

‖zI − Φp(θ)(θ)‖•,X ≥ c > 0 (5.5)

for θ ∈ Ω and z ∈ T. We postpone the proof of this claim. Combining (5.5), the

assumptions in part (a), and Lemma 5.6, we can verify the assumptions of Lemma 4.10.

But the conclusion of Lemma 4.10 contradicts (5.4) by virtue of the Discrete Dichotomy

Theorem.

To prove (5.5), suppose that inf‖zI −Φp(θ)(θ)‖•,X : θ ∈ Ω = 0 for some z ∈ T. Then

the conclusion of Lemma 4.8 hold, that is, there is a sequence of functions fn ∈ D(Γ)

given by (4.10) such that ‖fn‖∞ ≥ 1/2 and ‖(Γ− iξ)fn‖∞ → 0 for some ξ ∈ R, see (4.11).

Set gn = Γfn−iξfn and λ = iξ. In the next lemma, we show that fn = −Rλgn. But this is

impossible since Rλ is a bounded operator on Cb(Θ;X) due to condition (Mλ, Cb(Θ;X))

and Lemma 5.2.

Lemma 5.8. The functions fn defined in (4.10) and gn = (Γ − iξ)fn defined in (4.11)

satisfy fn = −Rλgn; that is, for t ≥ 0 and θ ∈ Θ one has

fn(ϕtθ)− Φtiξ(θ)fn(θ) = −

∫ t

0

Φt−τiξ (ϕtθ)gn(ϕτθ) dτ. (5.6)

Proof of Lemma 5.8. Recall that fn and gn are supported in Un = ϕsσ : 0 ≤ s <

p(σ), σ ∈ Σ, see (4.10) and (4.11). Fix θ = ϕsσ ∈ Un and t ≥ 0.

First, assume that s+ t < p(σ). Then, by (4.10),

fn(ϕtθ)− Φtiξ(θ)fn(θ) = fn(ϕt+sσ)− Φt

iξ(ϕsσ)fn(ϕsσ)

= β(σ)Φt+siξ (σ)

[α( t+s

p(σ)) (xn − Φ

p(σ)iξ xn) + Φ

p(σ)iξ (σ)xn

]− β(σ)Φt+s

iξ (σ)[α( s

p(σ)) (xn − Φ

p(σ)iξ xn) + Φ

p(σ)iξ (σ)xn

]= β(σ)Φt+s

iξ (σ)[α( t+s

p(σ))− α( s

p(σ))]

(xn − Φp(σ)iξ xn).

31

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On the other hand, (4.11) yields

−∫ t

0

Φt−τiξ (ϕτθ)gn(ϕτθ) dτ

= β(σ)

∫ t

0

Φt−τiξ (ϕτ+sσ)Φτ+s

iξ (σ)[xn − Φp(σ)iξ (σ)xn]

1

p(σ)α′( τ+s

p(σ)) dτ

= β(σ)Φt+siξ (σ)(xn − Φ

p(σ)iξ xn)

∫ t

0

α′( τ+sp(σ)

)dτ

p(σ),

and (5.6) holds.

Second, assume that s + t ∈ [kp(σ), (k + 1)p(σ)) for k = 1, 2, . . .. Now, ϕtθ = ϕt+sσ =

ϕt+s−kp(σ)σ, where t+ s− kp(σ) ∈ [0, p(σ)). The LHS of (5.6) can be written as

β(σ) Φt+s−kp(σ)iξ (σ)α( t+s−kp(σ)

p(σ)) [xn − Φ

p(σ)iξ (σ)xn] + β(σ)Φ

t+s−(k−1)p(σ)iξ (σ)xn

− β(σ) Φt+s+p(σ)iξ (σ)xn − β(σ) Φt+s

iξ (σ)α( sp(σ)

) [xn − Φp(σ)iξ (σ)xn].

We split the integral on the RHS of (5.6) in three integrals and compute

−∫ p(σ)−s

0

Φt−τiξ (ϕτθ)gn(ϕτθ) dτ

= β(σ)

∫ p(σ)−s

0

1p(σ)

α′( τ+sp(σ)

)Φt−τiξ (ϕs+τσ)Φτ+s

iξ (σ)[xn − Φp(σ)iξ (σ)xn] dτ

= β(σ)(

1− α( sp(σ)

))

Φt+siξ (σ)[xn − Φ

p(σ)iξ (σ)xn]

since α(1) = 1 by the choice of α. Further,

−∫ kp(σ)−s

p(σ)−sΦt−τiξ (ϕτθ)gn(ϕτθ) dτ

= β(σ)k−1∑`=1

∫ (`+1)p(σ)−s

`p(σ)−s

1p(σ)

α′( s+τ−`p(σ)p(σ)

) Φt−τiξ (ϕs+τ−`p(σ)σ)Φ

s+τ−`p(σ)iξ (σ)

· [xn − Φp(σ)iξ (σ)xn] dτ

= β(σ)[Φt+s−(k−1)p(σ)iξ (σ)xn − Φt+s

iξ (σ)xn],

where we have used α(0) = 0, α(1) = 1, and the cocycle property. Finally,

−∫ t

kp(σ)−sΦt−τiξ (ϕτθ)gn(ϕτθ) dτ

= β(σ)

∫ t

kp(σ)−s

1p(σ)

α′( τ+s−kp(σ)p(σ)

) Φt−τiξ (ϕτ+s−kp(σ)σ)Φ

τ+s−kp(σ)iξ (σ)[xn − Φ

p(σ)iξ (σ)xn] dτ

= β(σ) Φt+s−kp(σ)iξ (σ)α( t+s−kp(σ)

p(σ)) [xn − Φ

p(σ)iξ (σ)xn].

The lemma follows by combining these identities.

As a result, (5.3) is verified.

Step II. Next, we prove that I − πθ0(E) is invertible on c0(Z;X) for each θ0 ∈ Θ.

Together with (5.3), this implies (5.2) and so the theorem is established. We distinguish

between three cases.32

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(1) Assume that p(θ0) =∞. For γ ≥ 0, we define the operator

D(γ) : (xk) 7→ (xk − e−γ(|k|−|k−1|)Φ1(ϕk−1θ0)xk−1)

on c0(Z;X). Notice that

D(γ) → D(0) = I − πθ0(E) in L(c0(Z;X)) as γ → 0. (5.7)

By (5.3), this implies that there exist constants c, γ0 > 0 such that

‖D(γ)‖•,c0(Z;X) ≥ c for all 0 ≤ γ ≤ γ0. (5.8)

Take y = (yk) ∈ c0(Z;X) with yk = 0 for |k| ≥ n = n(y). Choose a function α ∈ C(R)

with suppα ⊂ (0, 1) and∫R α(τ)dτ = 1. For m = n+ 1, n+ 2, · · · and 0 ≤ γ ≤ γ0, define

gm(ϕtθ0) = α(t− k + 1) Φt−k+1(ϕk−1θ0) eγ|k−1| yk−1

for t ∈ [k − 1, k) and k = −m+ 1, . . . ,m. We can extend gm to a function gm ∈ C(Θ;X)

with compact support and

‖gm‖∞ ≤ ‖α‖∞ sup0≤τ≤1, θ∈Θ

‖Φτ (θ)‖ eγ0n ‖y‖∞ .

Condition (M0, Cb(Θ;X)) yields the function um = R0gm ∈ Cb(Θ;X), that is, one has

um(ϕkθ0) = Φ1(ϕk−1θ0)um(ϕk−1θ0) +

∫ 1

0

Φ1−τ (ϕτ+k−1θ0)g(ϕτ+k−1θ0) dτ

= Φ1(ϕk−1θ0)um(ϕk−1θ0) + Φ1(ϕk−1θ0)eγ|k−1|yk−1

for k = −m+ 1, . . . ,m. Set vm = (um(ϕkθ0) + eγ|k|yk)k∈Z ∈ `∞(Z;X) and

rm = (rmk)k∈Z = (I − πθ0(E))vm − (eγ|k|yk)k∈Z

for m > n. Then, (rmk)k∈Z ∈ `∞(Z;X) and rmk = 0 for |k| ≤ m. Since yk = 0 for |k| ≥ n,

we obtain

‖rm‖∞ ≤ ‖I − πθ0(E)‖L(`∞(Z;X)) ‖um‖∞ ≤ ‖I − πθ0(E)‖ ‖R0‖ ‖gm‖∞ ≤ C,

where C depends on y but not on m and γ ∈ [0, γ0]. Further, (e−γ|k|vmk)k∈Z ∈ c0(Z;X)

and

‖D(γ)(e−γ|k|vmk)k∈Z − y‖∞ = ‖(e−γ|k|[(I − πθ0(E))vm]k)k∈Z − y‖∞= sup

k∈Z|e−γ|k|rmk| ≤ C e−γm

for m > n. As a consequence, D(γ) has dense range on c0(Z;X) and is invertible by (5.8).

Now, (5.7) implies the invertibility of I − πθ0(E) on c0(Z;X) in case (1).

(2) Assume that p(θ0) ∈ (0,∞). We start with a special case.

(2i) Let p(θ0) = 1. Lemma 5.5 and (5.3) yield σap(Φ1(θ0))∩T = ∅. Further, eλ−Φ1(θ0)

is surjective by Lemma 5.6 for all λ = iξ ∈ iR in case (a) and for λ = 0 in case (b) of the

theorem. Therefore, eλ ∈ ρ(Φ1(θ0)) and the boundary of σ(Φ1(θ0)) does not intersect T.

As a result, Φ1(θ0) is hyperbolic and, by Lemma 5.5, I − πθ0(E) is invertible on c0(Z;X).

(2ii) Let p(θ0) = T ∈ (0,∞). We define the continuous flow ψt(θ) = ϕtT (θ) and the

exponentially bounded, strongly continuous cocycle Ψt(θ) = ΦtT (θ) over ψt on Θ. This

cocycle satisfies condition (MλT , Cb(Θ;X)). Also, for the flow ψt one has pψ(θ0) = 1. An

application of parts (I) and (II.2.i) to the closed ψt-invariant set O(θ0) and the cocycle33

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Ψ shows that the corresponding operators I − πθ(EΨ) are invertible on c0(Z;X) and

‖(I − πθ(EΨ))−1‖ ≤ C for θ ∈ O(θ0). Therefore Ψt has exponential dichotomy on O(θ0)

due to the Discrete Dichotomy Theorem 3.7. Hence, Φt has exponential dichotomy on

O(θ0) which implies the invertibility of I − πθ0(E).

(3) Assume that p(θ0) = 0. Then Φt(θ0) = etA is a strongly continuous semigroup

on X. For y ∈ X, choose g ∈ Cb(Θ;X) with g(θ0) = y. By condition (Mλ,F) (for all

λ = iξ ∈ iR in case (a) and for λ = 0 in case (b) of the theorem), there is a function

f ∈ Cb(Θ;X) such that f(θ0) =: x satisfies

eλtx− etAx =

∫ t

0

e(t−τ)Aeλτy dτ.

Consequently, x ∈ D(A) and (λ − A)x = y, i.e., λ /∈ σr(A). Lemma 5.5 and (5.3) yield

σap(eA)∩T = ∅. Hence, σap(A)∩ iR = ∅ by the spectral inclusion theorem. In particular,

λ /∈ σ(A) and the boundary of σ(A) does not intersect iR. So we infer that σ(A)∩ iR = ∅and, by the spectral mapping theorem for the residual spectrum, σr(e

A) ∩ T = ∅. As a

result, Φ1(θ0) is hyperbolic. So Lemma 5.5 implies 1 ∈ ρ(πθ0(E)).

We combine the results of this section in the following characterization.

Corollary 5.9. Let Φtt∈R+ be an exponentially bounded, strongly continuous cocycle

Φtt∈R+ over a continuous flow ϕtt∈R on a locally compact metric space Θ. Let F ∈Cb(Θ, X), C0(Θ, X).

(a) Assume that p(θ) ≥ d0 > 0 for θ ∈ B(θ). Then condition (Mλ,F) holds for all

λ = iξ ∈ iR if and only if Φt has exponential dichotomy on Θ.

(b) Assume that B(Θ) = ∅. Then condition (M0,F) holds if and only if Φt has

exponential dichotomy on Θ.

Note added in proof.

We remark that F. Rabiger [private communication] recently gave a direct proof of the fol-

lowing fact: If B(Θ) = ∅, then condition (M0, Cb(Θ;X)) implies condition (M0, C0(Θ;X)).

In other words, he proved that g ∈ C0(Θ;X) implies R0g ∈ C0(Θ;X) assuming condition

(M0, Cb(Θ;X)). The proof does not use operators πθ(E). It is based on a characterization

theorem for evolution semigroups and their generators similar to, e. g., Theorem 3.4 in

[38], and on the fact that Γu = −g for u, g ∈ C0(Θ;X) if and only if u and g satisfy the

mild integral equation (5.1) with λ = 0 (compare Lemma 1.1 in [32]). An indirect proof

of the fact above is contained in our Theorem 5.7.

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Y. Latushkin, Department of Mathematics, University of Missouri, Columbia, MO

65211, USA.

E-mail address: [email protected]

R. Schnaubelt, Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle

10, 72076 Tubingen, Germany.

E-mail address: [email protected]

36