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 C0 (Θ; X).
We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup 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 underlying 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 classical 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 {ϕt }t∈R be a continuous flow on a locally compact metric space Θ and let {Φt }t∈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
u̇(t) = A(ϕt θ)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
define the evolution semigroup {E t }t∈R+ by
(E t f )(θ) = Φt (ϕ−t θ)f (ϕ−t θ) for θ ∈ Θ, t ≥ 0, f ∈ F.
(1.1)
The semigroup {E t } 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 dim X < ∞
and compact Θ, the following Dichotomy Theorem is contained in [31]:
{Φt } has exponential dichotomy on Θ if and only if {E t } is hyperbolic;
that is, σ(E 1 ) ∩ T = ∅ for the spectrum σ(E 1 ) and the unit circle T. Moreover, in the
finite dimensional, compact setting it was proved in [7, 8, 9, 19] that the semigroup {E t }
satisfies the spectral mapping theorem
σ(E t )\{0} = exp(tσ(Γ))
for t ≥ 0
(1.2)
provided that the flow {ϕt } is aperiodic (i.e., Int{θ ∈ Θ : ϕT θ = θ for some T > 0} = ∅),
and the annular hull theorem
(σ(E t )\{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 Mañe Lemma, see also [7, 8, 9]. It says that 1 belongs to the approximate
point spectrum σap (E 1 ) if and only if there exists a point θ0 ∈ Θ, called the Mañe point,
such that sup{|Φt (θ0 )x0 | : t ∈ R} is finite for a nonzero vector x0 . We stress that,
in general, the Mañe 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 E 1 =: E can be expressed as a weighted translation operator E = aV on
F, where a(θ) = Φ1 (ϕ−1 θ) and (V f )(θ) = f (ϕ−1 θ). 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 x̄ = (xn )n∈Z
by the rule
πθ (E) : (xn )n∈Z 7→ (Φ1 (ϕn−1 θ)xn−1 )n∈Z = diag((a(ϕn θ))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 dimensional X, 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 ϕt θ = 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 which
2
can be thought as solution operator for the evolution equation u̇(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 investigated 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 {E t } 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
sup k[z − πθ (E)]−1 kL(c0 (Z;X)) < ∞.
z∈T,θ∈Θ
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=
∞
X
ak V
k
with kbk1 :=
k=−∞
∞
X
kak kL(C0 (Θ;X)) < ∞,
k=−∞
where ak are the multiplication operators induced by strongly continuous, bounded functions 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 {E t } 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 Section 3. Here, we replace the algebra B by a certain convolution algebra B̂. This allows
to prove the Discrete Dichotomy Theorem. In addition, for θ ∈ Θ, we introduce a family
of strongly continuous semigroups {Πtθ }t∈R+ on C0 (R; X) by setting
(Πtθ h)(s) = Φt (ϕs−t θ)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 u̇(t) = A(ϕt θ)u(t),
then the generator
d
·
t
Γθ of {Πθ } is the closure of the operator h 7→ − ds + 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 annular hull theorem provided that the function of prime periods is strictly separated from
3
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 dim X = ∞. 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 perturbation 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 construction 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 Mañe sequences {θm } ⊂ Θ which we define in this paper as an infinite dimensional
generalization of the Mañe points mentioned above. It is easy to see that σap (E) ∩ T 6= ∅
implies the existence of a Mañe sequence. As a generalization of the Mañe Lemma, we
show that the existence of a Mañe 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 uniqueness of bounded, continuous solutions to the mild inhomogeneous equation
Z t
t
t
u(ϕ θ) = Φ (θ)u(θ) +
Φt−τ (ϕτ θ)g(ϕt θ)dθ, t ≥ 0, θ ∈ Θ,
0
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 {ϕt }t∈R be a continuous flow on a locally compact
metric space Θ; that is, the map R × Θ 3 (t, θ) 7→ ϕt θ ∈ Θ is continuous and ϕt+τ θ =
ϕt (ϕτ θ), ϕ0 θ = θ for all t, τ ∈ R and θ ∈ Θ. Let {Φt }t≥0 be an exponentially bounded,
strongly continuous cocycle over {ϕt }t∈R . This means that the map R+ × Θ 3 (t, θ) 7→
Φt (θ) ∈ L(X) is strongly continuous, kΦt (θ)k ≤ N eωt for t ≥ 0, θ ∈ Θ, and some constants
N and ω, and the following cocycle property holds:
Φt+τ (θ) = Φt (ϕτ θ)Φτ (θ) and Φ0 (θ) = I
4
for t, τ ≥ 0 and θ ∈ Θ.
(2.1)
We will also consider cocycles for discrete time t ∈ N which are defined analogously.
Cocycles {Φt }t≥0 are in one-to-one correspondence with continuous linear skew-product
flows (LSPF) {ϕ̂t }t≥0 defined by
ϕ̂t : Θ × X → Θ × X : (θ, x) 7→ (ϕt θ, 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 {Φt }t≥0 has exponential dichotomy on Θ if there exists a
bounded, strongly continuous, projection valued function P : Θ → Ls (X) satisfying
(a) P (ϕt θ)Φt (θ) = Φt (θ)P (θ),
(b) ΦtQ (θ) is invertible as an operator from Im Q(θ) to Im Q(ϕt θ) (with inverse
t
Φ−t
Q (ϕ θ)),
(c) there exist constants β > 0 and M = M (β) ≥ 0 such that
kΦtP (θ)k ≤ M e−βt
and
k[ΦtQ (θ)]−1 k ≤ M e−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 ΦtQ (θ) denote the restrictions Φt (θ)P (θ) : Im P (θ) →
Im P (ϕt θ) and Φt (θ)Q(θ) : Im Q(θ) → Im Q(ϕt θ), 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 {Φt }t∈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 +t Φt (ϕτ θ0 ) = Φt (ϕτ θ0 )Pτ ,
(b) ΦtQ (ϕτ θ0 ) is invertible from Im Qτ to Im Qτ +t ,
(c) there exist constants β = β(θ0 ) > 0 and M = M (β) ≥ 0 such that
kΦtP (ϕτ θ0 )k ≤ M e−βt
and
k[ΦtQ (ϕτ θ0 )]−1 k ≤ M e−βt
for t, τ ∈ K and t ≥ 0.
Clearly, dichotomy on Θ implies dichotomy at θ0 . If Im Q(θ) and Im Q(ϕt θ) are finite
dimensional, then (b) and the second estimate in (c) follow from
|Φt (θ)x| ≥
1
M
eβt |x| for x ∈ Im Q(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
kΦt (θ0 )P [Φτ (θ0 )]−1 k ≤ M e−β(t−τ ) ,
t ≥ τ,
kΦt (θ0 )Q[Φτ (θ0 )]−1 k ≤ M e−β(τ −t) ,
t ≤ τ,
cf. [40]. Then one has Pτ = Φτ (θ)P0 [Φτ (θ)]−1 . This is the classical definition of exponential 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
Observe that the powers {E n }n∈N or a semigroup {E t }t≥0 have exponential dichotomy
if and only if E or E t0 for some t0 > 0 are hyperbolic, respectively. In this case, the
dichotomy projection is given by the Riesz projection
Z
1
P =
(λI − E)−1 dλ.
2πi T
We denote by EPt = E t P and EQt = E t Q the restriction to the stable and the unstable subspace, Im P and Im Q, 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−λt Φt (θ) and the rescaled linear skew-product flow
ϕ̂tλ : (θ, x) 7→ (ϕt θ, e−λt Φt (θ)x)
for θ ∈ Θ, x ∈ X, t ≥ 0.
Note that the cocycle Φtλ corresponds to the equation u̇(t) = [A(ϕt θ) − λ]u(t).
Definition 2.4. The dynamical (or Sacker-Sell) spectrum Σ is defined by
Σ = {λ ∈ R|
{Φtλ }t≥0
does not have exponential dichotomy}.
2.2. Evolution semigroups. By F = C0 (Θ; X) we denote the space of continuous functions f : Θ → X vanishing at infinity endowed with the sup-norm. For a cocycle {Φt }t≥0 ,
we introduce on C0 (Θ; X) the evolution semigroup {E t }t≥0 by setting
(E t f )(θ) = Φt (ϕ−t θ)f (ϕ−t θ)
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. {E t }t≥0 is a strongly continuous semigroup on C0 (Θ; X) if and only if
{Φt }t≥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 {Φt }t≥0 be an exponentially bounded,
strongly continuous cocycle over a continuous flow {ϕt }t∈R on Θ. If the induced evolution
semigroup {E t } is hyperbolic on C0 (Θ; X), then there exists a bounded, strongly continuous, projection valued function P : Θ → Ls (X) such that the Riesz projection P for E t
corresponding to σ(E t ) ∩ 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 {Φt }t≥0 be an exponentially bounded, strongly
continuous cocycle over a continuous flow {ϕt }t≥0 on a locally compact metric space Θ,
and let {E t }t≥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(E 1 ) ∩ D and the dichotomy
6
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 PE t = E t P by (a) in Definition 2.1. The inequality
kEPt kL(C0 (Θ;X)) ≤ M e−βt follows from the first inequality in (c). Also, EQt on Im Q has
the inverse (Rt f )(θ) = [ΦtQ (θ)]−1 f (ϕt θ). Now, (c) yields k[EQt ]−1 kL(C0 (Θ;X)) ≤ M e−βt for
t ≥ 0. This means that {E t } is hyperbolic.
(2) Assume that {E t } 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 E 1 , EQt is
invertible on Im Q and there exist constants β > 0 and M > 0 such that
kEPt kL(C0 (Θ;X)) ≤ M e−βt ,
t ≥ 0,
(2.2)
eβt kQf k∞ ,
t ≥ 0.
(2.3)
kEQt Qf k∞ ≥
1
M
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 kf k∞ = |x|. Using (2.2), we obtain
|ΦtP (θ)x| ≤ kΦtP (·)f (·)k∞ = kEPt f k∞ ≤ M e−βt kf k∞ = M e−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 kEQt f k∞ = supθ∈Θ |ΦtQ (θ)f (θ)|. For any > 0
choose f ∈ Im Q such that f (θ) = x and kEQt f k∞ ≤ |Φt (θ)x| + . 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
1 βt
e |x|
M
≤
1 βt
e kf k∞
M
≤ kEQt f k∞ ≤ |Φt (θ)x| + ε,
and hence
|ΦtQ (θ)x| ≥
1 βt
e |x|.
M
(2.4)
In particular, the operator ΦtQ (θ) : Im Q(θ) → Im Q(ϕt θ) is injective. To see that it is
surjective, take y ∈ Im Q(ϕt θ). For a continuous, compactly supported function α : Θ 7→
[0, 1] with α(ϕt θ) = 1, set f = α(·)Q(·)y. Then f ∈ Im Q. Since EQt is invertible on Im Q,
there exists g ∈ Im Q such that E t Qg = f For x := g(θ) ∈ Im Q(θ0 ), this gives
ΦtQ (θ)x = ΦtQ (θ)g(θ) = f (ϕt θ) = α(ϕt θ)Q(ϕt θ)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 {Φt }t≥0 the following assertions are equivalent:
(1) {Φt }t≥0 has an exponential dichotomy over R;
(2) the discrete time cocycle {Φt }t∈N has dichotomy.
7
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(E 1 ) \ {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
{Φt }t≥0 has exponential dichotomy. There exists ε > 0 such that every exponentially
bounded, strongly continuous cocycle {Ψt }t≥0 with
sup kΦt (θ) − Ψt (θ)kL(X) < ε
θ∈Θ
for some t > 0 also has exponential dichotomy.
Proof. Let {E t } and {F t } denote the evolution semigroups induced by {Φt } and {Ψt },
respectively. Assume that
sup kΦt (θ) − Ψt (θ)kL(X) < inf k(λ − E t )−1 k−1 =: ε.
λ∈T
θ∈Θ
Then, using λ − F t = [1 − (F t − E t )(λ − E t )−1 ] (λ − E t ) and
kE t − F t kL(C0 (Θ;X)) = sup kΦt (θ) − Ψt (θ)kL(X) ,
θ∈Θ
t
t
we derive that F , and hence {Ψ }, has exponential dichotomy.
3. Global and Pointwise Dichotomies
We now want to characterize the (global) dichotomy of {Φt } on Θ by the (local) dichotomy 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, equivalently, the hyperbolicity of {E t }) 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 {ϕt }t∈R on a locally compact metric space Θ, let us define the translation group (V t f )(θ) = f (ϕ−t θ) for t ∈ R
on the space C0 (Θ; X). Set at (θ) = Φt (ϕ−t θ) for t ≥ 0, θ ∈ Θ, and an exponentially
bounded, strongly continuous cocycle {Φt }t≥0 over {ϕt }t∈R . We abbreviate V = V 1 ,
E = E 1 , ϕ = ϕ1 , Φ = Φ1 .
We denote by à = Cb (Θ; Ls (X)) the algebra of bounded, strongly continuous, operator
valued functions a = a(·) on Θ with the sup-norm kakà = kak∞ = supθ∈Θ ka(θ)kL(X) and
point-wise multiplication. Clearly, Ã 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 B̂ denote the set of all sequences (ak )k∈Z with entries ak ∈ à satisfying (kak kà )k∈Z ∈
1
` (Z). Equipped with the norm
k(ak )k∈Z k1 :=
∞
X
kak kà =
k=−∞
∞
X
k=−∞
8
sup kak (θ)kL(X) ,
θ∈Θ
B̂ is a Banach space. For b̂0 = (a0n )n∈Z and b̂00 = (a00n )n∈Z in B̂, we define b̂ := b̂0 ∗ b̂00 by
P
b̂ = (an )n∈Z and an (θ) = k∈Z a0k (θ)a00n−k (ϕ−k θ) for θ ∈ Θ and n ∈ Z. Notice that
kan (θ)kL(X) ≤ kb̂0 k1 kb̂00 k1 .
The function θ 7→ a0k (θ)a00n−k (ϕ−k θ)x is continuous for x ∈ X and k, n ∈ Z. We have
X
|[an (θ1 ) − an (θ2 )]x| ≤
| [a0k (θ1 )a00n−k (ϕ−k θ1 ) − a0k (θ2 )a00n−k (ϕ−k θ2 )]x |
|k|≤N
+ 2 |x| kb̂00 k1
X
ka0k k∞ .
|k|>N
for θ1 , θ2 ∈ Θ and N ∈ N. Thus an ∈ Ã. We further estimate
X X
X X
kb̂k1 ≤
ka0k k∞ ka00n−k k∞ ≤ kb̂0 k1 kb̂00 k1 .
ka0k k∞ ka00n−k ◦ ϕ−k k∞ =
n
k
k
n
Therefore, (B̂, ∗, k · k1 ) is a Banach algebra with the unit element ê = e ⊗ δn0 . Here and
below, e(θ) = IX for θ ∈ Θ and δnk is the Kronecker delta. In other words, ê = (en ) where
e0 = I and en = 0 for n 6= 0.
To relate the algebra B̂ with L(C0 (Θ; X))), we introduce the (algebra) homomorphism
∞
X
ρ : B̂ → L(C0 (Θ; X))) : b̂ = (ak )k∈Z 7→
ak V k .
(3.1)
k=−∞
Clearly, ρ(ê) = I and kρ(b̂)kL(C0 (Θ;X)) ≤ kb̂k1 . We stress that, in general, ρ is not injective.
As an example, let ϕ1 θ = θ for all θ ∈ Θ. Then, V = I and each a ∈ A ⊂ L(C0 (θ; X))
can be represented as
a = 21 a + 12 aV = 31 a + 23 aV 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=
∞
X
ak V
k
where ak ∈ A and kbk1 :=
∞
X
kak kA < ∞.
(3.2)
k=−∞
k=−∞
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, k · k1 ) is a Banach algebra (with respect to the
composition of operators). Thus, ρ : B̂ → 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 =
∞
X
dk S k ∈ L(c0 (Z; X)) : dk ∈ D, kDkC :=
k=−∞
∞
X
kdk kL(c0 (Z;X)) < ∞},
k=−∞
where S : (xn )n∈Z 7→ (xn−1 )n∈Z is the shift operator on c0 (Z; X). The representation of
P
D ∈ C as D = dk S k is unique since, for m0 , n0 ∈ Z and x ∈ X with |x| = 1, one has
X
(n)
kDkL(c0 (Z;X)) ≥ kD(x ⊗ δn,n0 )kc0 (Z;X) = diag(dk )n∈Z (x ⊗ δn−k,n0 )n∈Z k
c0 (Z;X)
X (m )
(m )
≥|
dk 0 x δm0 −k,n0 | = |dm00−n0 x|.
k
9
We further need, for each θ ∈ Θ, the map
π̂θ :B̂ → L(c0 (Z; X)) : b̂ = (ak )n∈Z 7→
∞
X
diag[ak (ϕn θ)]n∈Z S k .
(3.3)
k=−∞
Observe that π̂θ : B̂ → C and
∞
X
kπ̂θ (b̂)kC =
k diag(ak (ϕn θ))n∈Z kL(c0 (Z;X))
=
k=−∞
∞
X
k=−∞
n
sup sup kak (ϕ θ)kL(X) =
n∈Z θ∈Θ
∞
X
kak k∞ = kb̂k1 .
k=−∞
Also, we compute for b̂ = b̂0 ∗ b̂00
π̂θ (b̂) =
∞
X
diag
k=−∞
=
∞
X
!
a0l (ϕn θ)a00k−l (ϕn−l θ)
l=−∞
n∈Z
"
∞
X
l=−∞
=
∞
X
diag(a0l (ϕn θ))n∈Z S l S −l
∞
X
diag(a0l (ϕn θ))n∈Z S l
l=−∞
0
#
diag(a00k−l (ϕn−l θ))n∈Z S k
k=−∞
"
∞
X
Sk
#
diag(a00k−l (ϕn θ))n∈Z S k−l
k=−∞
00
= π̂θ (b̂ ) · π̂θ (b̂ ).
As a result, π̂θ is a continuous homomorphism. Its importance relies on the following fact.
T
Proposition 3.1. θ∈Θ ker π̂θ = {0}.
Proof. Assume that π̂θ (b̂) = 0 for all θ ∈ Θ and some b̂ = (an )n∈Z ∈ B̂. This means that
X
0 = π̂θ (b̂) =
diag(ak (ϕn θ))n∈Z S k ∈ C
k
P
(n)
for θ ∈ Θ. Since the representation D = k diag(dk )S k for D ∈ C is unique, we conclude
that ak (ϕn θ) = 0 for each k, n ∈ Z and θ ∈ Θ. Thus, b̂ = 0 in B̂.
For a ∈ A, we define the operator b = I − aV ∈ L(C0 (Θ; X)) and the sequence
b̂ = (ak )k∈Z ∈ B̂ by a0 = e, a1 = −a, and ak = 0 otherwise. Then ρ(b̂) = 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(ϕn θ))n∈Z S.
(3.4)
To relate these concepts to the evolution semigroup {E t } on C0 (Θ; X) induced by the
cocycle {Φt }, we remark that the operator I − E 1 is of the form b = I − E = I − aV with
a(θ) = Φ(ϕ−1 θ) 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
k[πθ (b)]−1 kL(c0 (Z;X)) ≤ B
Then b−1 exists in L(C0 (Θ; X)).
10
for all
θ ∈ Θ.
(3.5)
Proof. Let b̂ = (ak )k∈Z ∈ B̂ be given by a0 = I, a1 = −a, and ak = 0 for k 6= 0, 1. The
definitions (3.3) and (3.4) yield
for θ ∈ Θ.
π̂θ (b̂) = πθ (b)
(3.6)
Also, ρ(b̂) = b by (3.1). Setting d0 (θ) = I, d1 (θ) = diag(a(ϕn θ))n∈Z , and dk (θ) = 0 for
k 6= 1, 2 and θ ∈ Θ, we see that
X
π̂θ (b̂) =
dk (θ)S k ∈ C.
k
Next, we show that the inverse of π̂θ (b̂) 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 [π̂θ (b̂)]−1 =
[πθ (b)]−1 belongs to C for each θ ∈ Θ; that is,
∞
X
(n)
[π̂θ (b̂)]−1 =
Ck (θ)S k where Ck (θ) = diag(Ck (θ))n∈Z satisfies
(3.7)
k=−∞
∞
X
k=−∞
sup kCk (θ)kL(c0 (Z;X)) < ∞.
(3.8)
θ∈Θ
Proof of Proposition 3.3. Set Dθ = π̂θ (b̂) = I − diag(a(ϕn θ))n∈Z S. Assumption (3.5) and
identity (3.6) yield
sup kDθ−1 k ≤ B.
(3.9)
θ∈Θ
For γ > 0, we define the operator
Dθ (γ) = I − diag(eγ(|n|−|n−1|) a(ϕn θ))n∈Z S
on c0 (Z; X). Note that Dθ (γ) = Jγ−1 Dθ Jγ for Jγ = diag(e−γ|n| )n∈Z . Clearly,
kDθ − Dθ (γ)kL(c0 (Z;X)) ≤ max{|1 − eγ |, |1 − e−γ |} kakA → 0
as γ → 0. Choose γ0 such that
q := B max{|1 − e±γ0 |} kakA < 1.
Then kDθ − Dθ (γ)k · kDθ−1 k ≤ q for all θ ∈ Θ and 0 ≤ γ ≤ γ0 . Since Dθ is invertible
B
in L(c0 (Z; X)), we conclude that Dθ (γ) is invertible and k[Dθ (γ)]−1 k ≤ R := 1−q
for all
0 ≤ γ ≤ γ0 and θ ∈ Θ.
Consider Dθ−1 as an operator matrix Dθ−1 = [Ckj (θ)]k,j∈Z , where Ckj (θ) ∈ L(X) are
defined for x ∈ X by
Ckj (θ)x := (Dθ−1 (x̄j ))k
for x̄j = (δjn x)n∈Z .
From Dθ (γ)−1 = Jγ−1 Dθ−1 Jγ we derive Dθ (γ)−1 = [eγ(|k|−|j|) Ckj (θ)]k,j∈Z . Now,
R ≥ kDθ (γ)−1 kL(c0 (Z;X)) ≥ eγ(|k|−|j|) kCkj (θ)kL(X)
for k, j ∈ Z yields
kCk,0 (θ)kL(X) ≤ R e−γ|k|
for θ ∈ Θ and k ∈ Z.
Observe that, for every j, k, m ∈ Z, one has
Dϕm θ = S −m Dθ S m
and Ckj (ϕm θ) = Ck+m,j+m (θ).
11
(3.10)
In particular, from Ck+m,m (θ) = Ck,0 (ϕm θ) it follows that
kCk+m,m (θ)kL(X) ≤ Re−γ|k|
for k, m ∈ Z and θ ∈ Θ.
(3.11)
So we can write Dθ−1 as
Dθ−1 =
X
k
Ck (θ)S k =
X
(n)
k
diag(Ck (θ))n∈Z S k
(n)
for Ck (θ) = Cn,n−k (θ), and we estimate
X
X
X
sup kCk (θ)kL(c0 (Z;X)) =
sup sup kCn,n−k (θ)kL(X) ≤
Re−γ|k| < ∞.
k∈Z
θ∈Θ
k∈Z
θ∈Θ n∈Z
k∈Z
To conclude the proof of Theorem 3.2 we need the following fact.
(0)
Claim 1. θ 7→ Ck (θ), see (3.7), is a continuous, bounded function from Θ to Ls (X)
with
X
(0)
sup kCk (θ)kL(X) < ∞.
k∈Z
θ∈Θ
We finish the proof of the theorem and show Claim 1 later. First, notice that (3.10)
implies
(n)
(0)
Ck (θ) = Cn,n−k (θ) = C0,−k (ϕn θ) = Ck (ϕn θ)
(3.12)
(0)
for k, n ∈ Z and θ ∈ Θ. The sequence dˆ = (Ck )k∈Z belongs to B̂ by Claim 1. Thus, we
ˆ We want to show
can apply to dˆ the homomorphism ρ : B̂ → L(C0 (Θ; X)). Let d = ρ(d).
that db = bd = I.
Consider the sequence r̂ = dˆ ∗ b̂ − ê ∈ B̂. Fix θ ∈ Θ and apply π̂θ : B̂ → C. Using
(3.3), (3.12), and Proposition 3.3, we compute
hX
i
(0)
ˆ · π̂θ (b̂) − I =
π̂θ (r̂) = π̂θ (d)
diag(Ck (ϕn θ))n∈Z S k · π̂θ (b̂) − I
k
hX
i
(n)
=
diag(Ck (θ))n∈Z S k · π̂θ (b̂) − I
k
hX
i
=
Ck (θ)S k · π̂θ (b̂) − I = 0.
k
Since θ ∈ Θ was arbitrary, Proposition 3.1 shows that r̂ = 0 in B̂. As a consequence,
ˆ · ρ(b̂) − I = d · b − I.
0 = ρ(r̂) = ρ(dˆ ∗ b̂ − ê) = ρ(d)
Similarly, one obtains 0 = bd − I, and the theorem is proved.
Proof of Claim 1. Let k ∈ Z, x ∈ X, and θ0 ∈ Θ. Define x̄ = (xn ) ∈ c0 (Z; X) by xn = x
if n = −k and xn = 0 if n 6= −k. Then, for every θ ∈ Θ,
∞
X
h
i
(0)
(0)
(0)
(0)
Cl (θ) − Cl (θ0 ) x−l [Ck (θ) − Ck (θ0 )]x = l=−∞
X h
i
(n)
(n)
≤ sup Cl (θ) − Cl (θ0 ) xn−l l
n∈Z
X
(n)
(n)
l
=
diag Cl (θ) − Cl (θ0 )
S x̄
l
n∈Z
c0 (Z;X)
= [π̂θ (b̂)]−1 − [π̂θ0 (b̂)]−1 x̄
c0 (Z;X)
≤ [π̂θ (b̂)]−1 · [π̂θ0 (b̂) − π̂θ (b̂)]ȳ ,
(3.13)
L(c0 (Z;X))
12
c0 (Z;X)
where ȳ := [π̂θ0 (b̂)]−1 x̄ ∈ c0 (Z; X). It is easy to see that
Θ 3 θ 7→ πθ (b) ∈ Ls (c0 (Z; X))
(0)
is (strongly!) continuous. So (3.13) and (3.9) imply the strong continuity of θ 7→ Ck (θ) ∈
Ls (X). ¿From (3.12) and (3.8) we derive
X
X
X
(0)
(0)
sup kCk (θ)kL(X) =
sup sup kCk (ϕn θ)kL(X) =
sup kCk (θ)kL(c0 (Z;X)) < ∞,
k∈Z
θ∈Θ
k∈Z
θ∈Θ n∈Z
k∈Z
θ∈Θ
and Claim 1 is proved.
Remark 3.4. Assume that ϕ is aperiodic. Then, ρ : B̂ → B is an isomorphism and
πθ = π̂θ ◦ 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 b̂ is
(0)
invertible in B̂ under condition (3.5). Its inverse is given by dˆ = (Ck )k∈Z . Later, dealing
with eventually norm-continuous in θ cocycles, we will need the following refinement of
this fact. Let Ãnorm be the closed subalgebra Cb (Θ; L(X)) of à of all norm continuous,
bounded, operator valued functions a : Θ → L(X). Let B̂norm denote the corresponding
closed subalgebra of B̂, that is, B̂norm = {(ak )k∈Z ∈ B̂ : ak ∈ Ãnorm }.
Corollary 3.5. Assume that a ∈ Cb (Θ; L(X)) is norm continuous. If the conditions of
Theorem 3.2 hold, then dˆ = b̂−1 belongs to B̂norm .
(0)
Proof. We have to check that θ 7→ Ck (θ) ∈ L(X) is norm continuous (cf. Claim 1 in
the proof of Theorem 3.2). Fix k ∈ Z. Recall the estimate (3.13)
(0)
(0)
|[Ck (θ) − Ck (θ0 )]x| ≤ B k[π̂θ0 (b̂) − π̂θ (b̂)]ȳkc0 (Z;X) ,
for x ∈ X and the constant B given in (3.5), and
−1
ȳ = [π̂θ0 (b̂)] x̄ =
∞
X
(n)
(n)
diag(Cl (θ0 ))n∈Z S l x̄ = (Cn−k (θ0 )x)n∈Z = (Cn,k (θ0 )x)n∈Z ,
l=−∞
where x̄ = δn,−k ⊗ x and we have used (3.12). The inequality (3.11) yields kCn,k (θ0 )k ≤
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
ka(ϕn θ) − a(ϕn θ0 )kL(X) < ε provided |n| < N and d(θ, θ0 ) < δ.
For d(θ, θ0 ) < δ, we now conclude that
k[π̂θ0 (b̂) − π̂θ (b̂)]ȳkc0 (Z;X)
≤ max{ sup |[a(ϕn θ) − a(ϕn θ0 )]Cn,k (θ0 )x|, sup |[a(ϕn θ) − a(ϕn θ0 )]Cn,k (θ0 )x|}
|n|<N
|n|≥N
≤ ε max{R, 2 sup ka(θ)kL(X) } |x|.
θ∈Θ
13
3.2. Discrete Dichotomy Theorem. The algebraic method developed above now enables 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−1 θ)f (ϕ−1 θ) on C0 (Θ; X), and
the hyperbolicity of the operators πθ (E) = diag(Φ(ϕn−1 θ))n∈Z S on c0 (Z; X).
We start with some elementary facts. First, observe that
πθ (E)k = πθ (E k ) = diag(Φk (ϕn−k θ))n∈Z S k
(3.14)
for k ∈ N and θ ∈ Θ. We set kT k•,Y = inf{kT yk : kyk = 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 kI − πθ (E)k• = kzI − πθ (E)k•
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
k(I − πθ (E))−1 k = k(zI − πθ (E))−1 k.
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
−iξ
Lξ [zI − πθ (E)] L−1
diag(Φ(ϕn−1 θ))n∈Z S = e−iξ (zeiξ I − πθ (E))
ξ = zI − 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 {Φt }t∈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 constants β(θ) ≥ β > 0 and M (β(θ)) ≤ M .
(c) πθ (E) is hyperbolic on c0 (Z; X) for each θ ∈ Θ and
sup sup k[zI − πθ (E)]−1 kL(c0 (Z;X)) < ∞.
z∈T θ∈Θ
(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 (ϕn θ),
and
pθ = diag(Pn )n∈Z .
Proof. (a) ⇒ (b). Let {Φt }t≥0 have exponential dichotomy for discrete time with projections P (·) ∈ Cb (Θ; Ls (X)) and constants β, M > 0, see Definition 2.1. For θ ∈ Θ, the
projections Pn := P (ϕn θ), n ∈ Z, clearly satisfy Definition 2.2 with constants β(θ) = β
and M (β(θ)) = M .
14
(b) ⇒ (c) Fix θ ∈ Θ. Let Pn , n ∈ Z, be the family of dichotomy projections for {Φt }
at θ as in Definition 2.2. Define pθ = diag(Pn )n∈Z on c0 (Z; X). By (a) in Definition 2.2,
we have pθ πθ (E) = πθ (E)pθ . The identity (3.14) and Definition 2.2(b) imply that
(πθ (E)pθ )k = diag(Φk (ϕn−k θ)Pn−k )n∈Z S k
and
n+k
θ)Qn+k )n∈Z S −k ,
(πθ (E)qθ )−k = diag([ΦkQ (ϕn θ)Qn ]−1 )n∈Z S −k = diag(Φ−k
Q (ϕ
for k ∈ N. Thus, the estimates in Definition 2.2(c) give
k(πθ (E)pθ )k kL(c0 (Z;X)) ≤ M e−βk
and k(πθ (E)qθ )−k kL(c0 (Z;X)) ≤ M e−βk .
As a result, πθ (E) is hyperbolic and (z, θ) 7→ k[zI − πθ (E)]−1 kL(c0 (Z;X)) is bounded.
(c) ⇒ (d). For every z ∈ T, apply Theorem 3.2 to b = I −aV , where a(θ) = z −1 Φ(ϕ−1 θ)
and (V f )(θ) = f (ϕ−1 θ) 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 {Φt }t∈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 pθ χ = 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 pθ πθ (E) = πθ (E)pθ
we derive (a) in Definition 2.2. Further, Φk (ϕn−k θ) : Im Qn−k → Im Qn 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, {Πtθ }t≥0 on C0 (R; X), which are
defined by
(Πtθ h)(s) = Φt (ϕs−t θ)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 (ϕτ θ), s ≥ τ , yield an exponentially
bounded, strongly continuous evolution family, see [21, p.110]. In particular, (Πtθ h)(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 {Πtθ }t≥0 is hyperbolic if and only if Γθ is
invertible. But dichotomy of {Uθ (s, τ )}s≥τ means exactly the dichotomy of {Φt }t≥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 {Φt }t≥0 over a continuous flow {ϕt }t∈R on Θ.
(a) {Φt }t≥0 has exponential dichotomy on Θ.
(b) σ(Π1θ ) ∩ T = ∅ for each θ ∈ Θ and
sup sup k[z − Π1θ ]−1 kL(C0 (R;X)) < ∞.
z∈T θ∈Θ
15
(c) σ(Γθ ) ∩ iR = ∅ for each θ ∈ Θ and
sup sup k[iξ − Γθ ]−1 kL(C0 (R;X)) < ∞.
ξ∈R θ∈Θ
(d) {Φt }t≥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 {Pτθ }τ ∈R for the dichotomy of {Φt } at a point θ ∈ Θ are related as
follows:
P = P (·), Pτθ = P (ϕτ θ), (pθ h)(τ ) = Pτθ h(τ ), h ∈ C0 (R; X).
3.3. Perturbation theorem. In this subsection we give an infinite-dimensional generalization 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 {Φn }n∈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 (ϕn θ))n∈Z . Notice
that a = Φ1 ◦ ϕ−1 ∈ Ãnorm , see the notations before Corollary 3.5. Define â = (an )n∈Z ∈
B̂norm by a1 = a and an = 0 otherwise. Let b̂z = ê − z −1 â for z ∈ T. Recall that
π̂θ (b̂z ) = πθ (I − z −1 aV ) = I − z −1 πθ (E) by (3.6). The Discrete Dichotomy Theorem 3.7
yields
k(I − z −1 πθ (E))−1 kL(c0 (Z;X)) ≤ C for all θ ∈ Θ and z ∈ T.
Applying Corollary 3.5 to bz = I − z −1 aV , we see that for each z ∈ T the element
zê − â ∈ B̂norm is invertible in B̂norm . Define the idempotent p̂ in the algebra B̂norm by
Z
1
p̂ =
(zê − â)−1 dz.
2πi T
Since π̂θ : B̂ → C is a continuous homomorphism for all θ ∈ Θ, we obtain
Z
Z
1
1
−1
π̂θ ([zê − â] ) dz =
[zI − πθ (aV )]−1 dz = pθ
π̂θ (p̂) =
2πi T
2πi T
for every θ ∈ Θ. On the order hand, let p̂ = (Pk )k∈Z for Pk ∈ Ãnorm . Then we have
X
πθ (Pk )S k
0 = π̂θ (p̂) − pθ = πθ (P0 ) − pθ +
k6=0
by the definition (3.3) of π̂θ and πθ (Pk ) = diag[Pk (ϕn θ)]n . Since the representation of
π̂θ (p̂) − 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 ∈ Ânorm (use the sequence (δn0 Pk )n ∈ B̂).
16
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 ) = sup inf d(θ0 , θ1 ) for Θ1 ∈ K.
θ1 ∈Θ1 θ0 ∈Θ0
Further, denote by Σ(Θ1 ) = Σ({ϕ̂n |Θ1 × X}) the Sacker-Sell dynamical spectrum, see
Definition 2.4, for the cocycle {Φn }n∈N restricted on Θ1 , that is, for {Φn }n∈N = {Φn (θ) :
n ∈ N, θ ∈ Θ1 }, or for the linear skew-product flow {ϕ̂n } 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 {Φt }t∈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 {Φt }t∈N has exponential dichotomy on the set Θ0 ∈ K in the sense of Definition
2.1. There exists δ∗ > 0 such that the cocycle {Φt }t∈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
θ̄ = {θn }n∈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 Tθ̄ : (xn )n∈Z 7→ (Φ(θn−1 )xn−1 )n∈Z
on c0 (Z; X) for every ε-pseudo-orbit θ̄ = {θn }n∈Z ⊂ Θ0 .
Assuming that {Φt } has exponential dichotomy on Θ0 , we will show that for sufficiently
small ε the operators I − Tθ̄ are invertible on c0 (Z, X) with uniformly bounded norms
k(I − Tθ̄ )−1 k for all ε-pseudo-orbits θ̄ ⊂ Θ0 . Fix θ ∈ Θ1 and choose an ε-pseudo-orbit
θ̄ = {θn } ⊂ Θ0 such that d(θn , ϕn θ) is small for each n ∈ Z. We use the continuity of Φ(·)
to show that for this choice of {θn } the norm kπθ (E) − Tθ̄ kL(c0 (Z;X)) is small for sufficiently
small ε > 0. Since I − Tθ̄ is invertible for θ̄ ⊂ Θ0 , we conclude that I − πθ (E) is invertible
and k(I − πθ (E))−1 k ≤ 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 {kP (θ)k, 1}. Choose γ > 0 such that
γ≤
1
.
16c
(3.16)
Fix N ∈ N such that M e−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) : Im Q(θ) → Im Q(ψθ) is invertible,
(3.18)
kΨP (θ)k ≤ γ,
(3.19)
kΨ−1
Q (θ)k ≤ γ.
(3.20)
17
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 kΨ(θ) − Ψ(η)k ≤ 3/16,
supd(θ,η)≤ε, θ,η∈Θ0 kP (θ) − P (η)k ≤
1
,
2(1+2c)
c supd(θ,η)≤ε, θ,η∈Θ0 kP (θ) − P (η)k · maxθ∈Θ kΨ(θ)k ≤ 3/8.
(3.21)
(3.22)
(3.23)
For a sequence θ̄ = {θn }n∈Z ⊂ Θ0 , we define by
Tθ̄ : (xn ) 7→ (Ψ(θn−1 )xn−1 ), Pθ̄ = diag {P (θn )}, Qθ̄ = I − Pθ̄ , p̂θ̄ = diag {P (ψθn−1 )}
operators on c0 (Z; X). Then (3.17) implies
Tθ̄ Pθ̄ = p̂θ̄ Tθ̄
and Tθ̄ Qθ̄ = (I − p̂θ̄ )Tθ̄ .
(3.24)
The following lemma shows that exponential dichotomy, (3.17)–(3.20), implies the invertibility of I −Tθ̄ for every ε-pseudo-orbit θ̄ for ψ in Θ0 , where ε is given by (3.21)–(3.23).
Lemma 3.12. Let θ̄ = {θn } ⊂ Θ0 be an ε-pseudo-orbit for ψ, where ε satisfies (3.21)–
(3.23). Then I − Tθ̄ is invertible on c0 (Z, X) and
k(I − Tθ̄ )−1 k ≤ 38 ·
(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 (E N ) : (xn ) 7→
(Ψ(ψ n−1 θ)xn−1 ).
For a given θ ∈ Θ1 and each n ∈ Z, we use (3.15) to find θn ∈ Θ0 such that d(θn , ψ n θ) <
2δ. The sequence θ̄ = {θn } ⊂ Θ0 is an ε-pseudo-orbit for ψ because of
d(θn , ψθn−1 ) ≤ d(θn , ψ n θ) + d(ψ(ψ n−1 θ), ψ(θn−1 )) ≤ 2δ + ε/2 ≤ ε.
Thus, Lemma 3.12 can be applied to I − Tθ̄ yielding k(I − Tθ̄ )−1 k ≤ 8/3. Also,
kTθ − Tθ̄ k = sup kΨ(ψ n θ) − Ψ(θn )k ≤ sup kΨ(ϑ) − Ψ(η)k ≤ 3/16
n∈Z
(3.26)
d(ϑ,η)≤ε
by (3.21). Then the operator
I − Tθ = I − Tθ̄ − (Tθ − Tθ̄ ) = (I − Tθ̄ ) I − (I − Tθ̄ )−1 (Tθ − Tθ̄ )
is invertible by (3.25) and (3.26). Moreover, the usual estimate of the Neumann series
implies k(I − Tθ )−1 k ≤ 16/3.
As a consequence of the Discrete Dichotomy Theorem, E N 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) = Im Pθ̄ ⊕ Im Qθ̄ , we represent Tθ̄
as Tθ̄ = A + B by
Pθ̄ Tθ̄ Pθ̄
0
0
Pθ̄ Tθ̄ Qθ̄
A=
and B =
.
(3.27)
0
Qθ̄ Tθ̄ Qθ̄
Qθ̄ Tθ̄ Pθ̄
0
We prove the lemma in four steps.
18
Step 1. We show that
max{kPθ̄ Tθ̄ Pθ̄ k, k [Qθ̄ Tθ̄ Qθ̄ ]−1 k} ≤ 1/8.
(3.28)
First, (3.16) and (3.19) imply
kPθ̄ Tθ̄ Pθ̄ k = sup kP (θn )Ψ(θn−1 )P (θn−1 )k ≤ cγ ≤ 1/16 .
n∈Z
Further, by (3.24) one has
Qθ̄ Tθ̄ Qθ̄ = Qθ̄ (I − p̂θ̄ ) · (I − p̂θ̄ )Tθ̄ Qθ̄ .
(3.29)
The operator (I − p̂θ̄ )Tθ̄ Qθ̄ : ImQθ̄ → Im (I − p̂θ̄ ) is invertible by (3.18) and (3.20), and
k [(I − p̂θ̄ )Tθ̄ Qθ̄ ]−1 k = sup kΨ−1
Q (θn )k ≤ γ.
(3.30)
n∈Z
To prove that Qθ̄ (I − p̂θ̄ ) : Im (I − p̂θ̄ ) → Im Qθ̄ is invertible, we consider the operator
R := I − (Pθ̄ − Qθ̄ )(Pθ̄ − p̂θ̄ ) = Qθ̄ (I − p̂θ̄ ) + Pθ̄ p̂θ̄
on c0 (Z, X) and its decomposition
R : Im p̂θ̄ ⊕ Im (I − p̂θ̄ ) → Im Pθ̄ ⊕ Im Qθ̄ ;
Pθ̄
Pθ̄ p̂θ̄
0
R=
R [p̂θ̄ , I − p̂θ̄ ] =
.
Qθ̄
0
Qθ̄ (I − p̂θ̄ )
Since θ̄ = {θn } is an ε-pseudo-orbit for ψ, the inequality (3.22) yields
k(Pθ̄ − Qθ̄ )(Pθ̄ − p̂θ̄ )k ≤ (1 + 2c)kPθ̄ − p̂θ̄ k = (1 + 2c) sup kP (θn ) − P (ψθn−1 )k ≤ 1/2.
n∈Z
Therefore, R is invertible and, hence, Qθ̄ (I − p̂θ̄ ) is invertible with k[Qθ̄ (I − p̂θ̄ )]−1 k ≤
kR−1 k ≤ 2. This estimate together with (3.29)–(3.30) and (3.16) gives k [Qθ̄ Tθ̄ Qθ̄ ]−1 k ≤
2γ ≤ 1/8.
Step 2. We show that I − A is invertible, and
k(I − A)−1 k ≤ 4/3 .
(3.31)
The inequality (3.28) implies that the operators
Pθ̄ − Pθ̄ Tθ̄ Pθ̄
and Qθ̄ − Qθ̄ Tθ̄ Qθ̄ = −Qθ̄ Tθ̄ Qθ̄ Qθ̄ − (Qθ̄ Tθ̄ Qθ̄ )−1
are invertible on Im Pθ̄ and Im Qθ̄ , respectively. Then
(I − A)
−1
−1
= (Pθ̄ − Pθ̄ Tθ̄ Pθ̄ )
−1
⊕ (Qθ̄ − Qθ̄ Tθ̄ Qθ̄ )
=
∞
X
k
(Pθ̄ Tθ̄ Pθ̄ ) ⊕ −
k=0
∞
X
(Qθ̄ Tθ̄ Qθ̄ )−k .
k=1
−1
As a result, k(I − A) k ≤ 4/3.
Step 3. We estimate kBk = max{kPθ̄ Tθ̄ Qθ̄ k, kQθ̄ Tθ̄ Pθ̄ k}. By (3.24),
Pθ̄ Tθ̄ Qθ̄ = Pθ̄ (Pθ̄ − p̂θ̄ )Tθ̄
and Qθ̄ Tθ̄ Pθ̄ = (p̂θ̄ − Pθ̄ )p̂θ̄ Tθ̄ .
We also have kPθ̄ k ≤ c, kp̂θ̄ k ≤ c, kp̂θ̄ − Pθ̄ k ≤ supd(θ,η)≤ε kP (θ) − P (η)k, kTθ̄ k ≤
supθ∈Θ kΨ(θ)k. Thus, (3.23) yields
kBk ≤ c sup kP (θ) − P (η)k · sup kΨ(θ)k ≤ 3/8 .
θ∈Θ
d(θ,η)≤ε
19
(3.32)
Step 4. We use (3.31) and (3.32) to finish the proof. Indeed, k(I − A)−1 Bk ≤
1
2
and
k(I − Tθ̄ )−1 k = k[(I − A)(I − (I − A)−1 B]−1 k
≤ k(I − A)−1 k k[I − (I − A)−1 B]−1 k ≤ 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
u̇(t) = [A + B(ϕt θ)]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
sup kB(ϕt θ1 ) − B(ϕt θ2 )kL(X) → 0 as θ1 → θ2
t∈K
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 = etA x +
Zt
e(t−τ )A B(ϕτ θ)Φτ (θ)x dτ,
x ∈ X,
0
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 {Φt }t≥0 by the
spectrum of the generator Γ of the corresponding evolution semigroup {E t }t≥0 on the
space F = C0 (Θ; X). To that purpose, we prove the spectral mapping and annular hull
theorems for {E t }t≥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 Mañe sequences for {Φn }n∈Z+ . Throughout, {Φt }t≥0 is an
exponentially bounded, strongly continuous cocycle over a continuous flow {ϕt }t∈R on a
locally compact metric space Θ.
Definition 4.1. A sequence {θm }∞
m=1 ⊂ Θ is called Mañe sequence if there exist constants
C, c > 0 and vectors xm ∈ X such that
|Φm (θm )xm | ≥ c
and
|Φk (θm )xm | ≤ C
for
20
(4.1)
k = 0, 1, . . . , 2m.
(4.2)
We first discuss this concept, see also Corollary 4.12 and 4.13. Assume, for a moment,
that {Φn }n∈Z is a cocycle with invertible values and that θ0 is a Mañe point, i.e., there
exists a nonzero vector x0 such that C := sup{|Φn (θ0 )x0 | : n ∈ Z} < ∞, see the Introduction. We claim that θm = ϕ−m θ0 form a Mañe sequence with vectors xm = Φ−m (θ0 )x0 .
Indeed,
|Φm (θm )xm | = |Φm (ϕ−m θ0 )Φ−m (θ0 )x0 | = |x0 | =: c > 0 and
|Φk (θm )xm | = |Φk (ϕ−m θ0 )Φ−m (θ0 )x0 | = |Φk−m (θ0 )x0 | ≤ C
for k ∈ Z. Thus, Definition 4.1 extends the notion of a Mañe 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 {Φt }t∈N has exponential dichotomy on Θ, then there are no Mañe
sequences in Θ.
Proof. Suppose, in the contrary, that {θm }∞
1 is a Mañe sequence with the corresponding
s
u
vectors xm . Decompose xm = xm + xm , where xsm ∈ Im P (θm ) and xum ∈ Im Q(θm ). Let
s
u
ym = Φm (θm )xm , ym
= Φm (θm )xsm , and ym
= Φm (θm )xum . Using dichotomy and (4.2) with
k = 2m and k = m, we estimate
u
m
u
−βm
m
s
|ym
| ≤ M e−βm |Φm
{|Φ2m (θm )xm | + |Φm
Q (ϕ θm )ym | ≤ M e
P (ϕ θm )ym |}
u
u
≤ M e−βm {C + M e−βm (|ym | + |ym
|)} ≤ M e−βm {C + M e−βm (C + |ym
|)}.
Thus, for sufficiently large m, we have
C + M Ce−βm
→ 0 as m → ∞.
1 − M 2 e−2βm
s
| ∈ [c/2, 2C] for large m. Since
Together with (4.1) and (4.2) with k = m, this implies |ym
t
Φ has exponential dichotomy, we conclude
c
s
u
≤ |ym
| = |Φm (θm )xsm | ≤ M e−βm |xsm | and 2C ≥ |ym
| = |Φm (θm )xum | ≥ M −1 eβm |xum |.
2
Hence, |xsm | → ∞ and |xum | → 0 as m → ∞. This contradicts (4.2) with k = 0.
u
|ym
| ≤ M e−βm
We now give sufficient conditions for the existence of a Mañe 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
kI − πθ (E)k•,`∞ (Z;X) ≥ c for all θ ∈ Θ.
Then we have kzI − Ek•,F ≥ c for all z ∈ T.
Proof. Let f ∈ F and z ∈ T. Notice that for each θ ∈ Θ the sequence x̄θ := (f (ϕk θ))k∈Z
is bounded. The assumption and Proposition 3.6 yield
kzf − Ef kF = sup |zf (θ) − Φ1 (ϕ−1 θ)f (ϕ−1 θ)| = sup sup |zf (ϕk θ) − Φ1 (ϕk−1 θ)f (ϕk−1 θ)|
θ∈Θ
θ∈Θ k∈Z
= sup k[zI − πθ (E)]x̄θ k`∞ (Z;X) ≥ c sup kx̄θ k`∞ (Z;X) = c kf kF .
θ∈Θ
θ∈Θ
21
Proposition 4.4. A Mañe sequence {θm }∞
1 exists provided one of the following conditions
holds.
(a) inf θ∈Θ kI − πθ (E)k•,`∞ = 0.
(b) σap (E) ∩ T 6= ∅.
Proof. In view of Lemma 4.3, it suffices to consider (a). Since {Φt } is exponentially
P
k −1
bounded, N := supθ kπθ (E)k is finite. Fix m ∈ N and let ε := (4 2m
k=0 N ) . By the
(m)
assumption, there exists σm ∈ Θ and ȳ (m) = (yl )l∈Z ∈ `∞ (Z; X) such that kȳ (m) k∞ = 1
and k[I − πσm (E)]ȳ (m) k∞ < ε. Hence,
1
2
< k[πσm (E)]k ȳ (m) k`∞ (Z;X) < 2 for k = 0, 1, . . . 2m.
For a fixed k ∈ {0, 1, . . . , 2m}, we have
(m)
(m)
k[πσm (E)]k ȳ (m) kl∞ (Z;X) = sup |Φk (ϕl−k σm )yl−k | = sup |Φk (ϕl−m σm )yl−m |.
l∈Z
l∈Z
(m)
(m)
Select l such that 21 ≤ |Φm (ϕl−m σm )yl−m |. Note that |Φk (ϕl−m σm )yl−m | ≤ 2 holds for
k = 0, 1, . . . 2m. Therefore, θm := ϕl−m σm is a Mañe sequence with the vectors xm :=
(m)
yl−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 {ϕt }t∈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) = inf{T > 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) = {ϕt θ : t ∈ R} the orbit through θ ∈ Θ. The following result
is taken from [6, Thm. IV.2.11].
Lemma 4.5. Let {ϕt }t∈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−t θ 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 {ϕt }t∈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 U
and 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−t θ ∈ Σ, and U 3 θ 7→ t(θ) is continuous.
22
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 ϕt θ0 for some 0 ≤ t < p(θ0 ) such that O(θn )∩Σ1 is empty or contains more than one
element. But, since ϕ−t θn → θ0 , we obtain ϕ−t θn ∈ U0 and, hence, points σn ∈ O(θn ) ∩ Σ0
for large n. Moreover, t(ϕ−t θn ) → t(θ0 ) = 0 so that σn → θ0 . In particular, σn ∈ Σ1 for
large n. Then there must exist σn0 = ϕsn σn ∈ Σ1 for some sn with
τ
0 < τ ≤ sn ≤ p(σn ) − τ < p(θ0 ) −
2
and n sufficiently large, where we have used the continuity of p at θ0 . Taking a subsequence, we may assume that sn → s so that
σn0 → ϕs θ0 ∈ Σ1 ⊂ U0 .
Again by the continuity of t(·), we derive ϕs θ0 ∈ Σ0 . Due to (4.3), this yields ϕs θ0 = 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 → ϕp0 θ. Hence, p0 = kp(θ) for k 6= 1. This is impossible since p(θ), p0 ∈
[3p(θ0 )/4, 5p(θ0 )/4]. Further, ϕr θn → ϕr θ 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 {E t }t≥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 Mañe sequence. We first treat the case that the
Mañe sequence consists of aperiodic points.
Lemma 4.7. Assume {θm }∞
/ B(Θ) or p(θm ) →
1 is a Mañe sequence such that θm ∈
∞. Then, for each ξ ∈ R, there exist functions fn ∈ D(Γ) with kfn k∞ = 1 such that
k(iξ − Γ)fn k∞ → 0 as n → ∞.
Proof. Without
1 loss of generality, we assume that p(θm ) → ∞ as m → ∞. Denote
n = min{m, 5 p(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−n σ0 )x| and |Φk (ϕ−n σ0 )x| ≤ C
23
for k = 0, 1, . . . , 2n.
(4.5)
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−n σ)x − Φt (ϕ−n σ0 )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 ∈ C 1 (R) with 0 ≤ γ ≤ 1, supp γ ⊂ (0, 2n), γ(n) = 1, and kγ 0 k∞ ≤ 2/n.
For ξ ∈ R, define
fn (θ) = e−iξt β(σ)γ(t)Φt (ϕ−n σ)x if θ = ϕt−n σ ∈ U with t ∈ (0, 2n), σ ∈ Σ,
(4.7)
and fn (θ) = 0 if θ ∈
/ U. Clearly, fn ∈ F and kfn k∞ ≥ |fn (σ0 )| ≥ c > 0 by (4.5). For
h > 0 so small that γ = 0 on [2n − h, 2n], one has
(E h fn )(θ) = e−iξ(t−h) β(σ)γ(t − h)Φt (ϕ−n σ)x
for θ = ϕt−n σ ∈ U and (E h fn )(θ) = 0 for θ ∈
/ U. Thus, fn ∈ D(Γ) and
(Γfn )(θ) − iξfn (θ) = −γ 0 (t)β(σ)e−iξt Φt (ϕ−n σ)x
(4.8)
for θ = ϕt−n σ with t ∈ (0, 2n) and σ ∈ Σ and (Γfn )(θ) = 0 for θ ∈
/ U. Let N =
τ
sup{kΦ (θ)k : τ ∈ [0, 1], θ ∈ Θ}. Then (4.6) and (4.5) imply
kΓfn − iξfn k∞ ≤
2N
2N (C + 1)
max |Φk (ϕ−n σ)x| ≤
.
n k=0,...,2n
n
If the Mañe 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 kfn k∞ = 1 and k(iξ − Γ)fn k∞ → 0 as
n → ∞.
η
Proof. We may assume that z = eiη = ei 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 θn
such 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 ∈ C 1 [0, 1] satisfying 0 ≤ α ≤ 1, α = 0 on [0, 1/3], α = 1 on [2/3, 1],
and kα0 k∞ ≤ 4. Define
fn (θ) = e−iξt β(σ)Φt (σ)[α(t/p(σ))xn + (1 − α(t/p(σ)))e−iξp(σ) Φp(σ) (σ)xn ]
(4.10)
for θ = ϕt σ ∈ Un , where σ ∈ Σn and 0 ≤ t < p(σ), and fn (θ) = 0 for θ ∈
/ Un . Clearly,
fn ∈ F with
kfn k∞ ≥ |fn (θn )| = |Φp(θn ) (θn )xn | ≥ 1/2
for large n. For 0 < h < d0 /6, we compute
(
t−h
t−h
e−iξ(t−h) β(σ)Φt (σ)[α( p(σ)
)xn + (1 − α( p(σ)
))e−iξp(σ) Φp(σ) (σ)xn ], t ≥ h,
(E h fn )(θ) =
e−iξ(t−h) β(σ)Φt (σ)e−iξp(σ) Φp(σ) (σ)xn ,
t < h,
24
for θ = ϕt σ ∈ Un and (E h fn )(θ) = 0 otherwise. (Write ϕ−h θ = ϕp(θ)+t−h σ if θ ∈ Un and
t − h < 0.) Thus, fn ∈ D(Γ) and
Γfn (θ) − iξfn (θ) = −e−iξt β(σ)
1 0 t
α(
)Φt (σ)[xn − e−iξp(σ) Φp(σ) (σ)xn ]
p(σ) p(σ)
(4.11)
for θ = ϕt σ ∈ Un and zero otherwise. Let N = sup{kΦτ (θ)k : θ ∈ Θ, 0 ≤ τ ≤ 2d1 }.
Finally we obtain
k(Γ − iξ)fn k∞ ≤
8N
|xn − e−iξp(σ) Φp(σ) (σ)xn | → 0
d0
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
(E t f )(θ) = Φt (ϕ−t θ)f (ϕ−t θ) on F = C0 (Θ; X)
with generator Γ. The spectral mapping theorem says that
σ(E t )\{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(E t )\{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(E t )\{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 {E t }t≥0 be the evolution semigroup on F = C0 (Θ; X) with generator Γ induced by an exponentially bounded,
strongly continuous cocycle {Φt }t∈R+ over a continuous flow {ϕt }t∈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(E t )
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
for E = E 1 . So let z ∈ σap (E) ∩ T. By Proposition 4.4, there exists a Mañe 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 {ψ t }t∈R by ψ t (θ) = ϕtp(θ) (θ) and a new cocycle
{Ψt }t∈R+ over {ψ t } by Ψt (θ) = Φtp(θ) (θ). Recall that θ 7→ p(θ) is continuous on BC(Θ).
Clearly, {ψ t } is continuous and {Ψt } is strongly continuous and exponentially bounded
on Ω. So there exists the induced evolution semigroup {EΨt } on C0 (Ω; X). Let EΨ = EΨ1 .
We stress that (EΨ f )(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
sup k[zI − Φp(θ) (θ)]−1 kL(X) < ∞
for all z ∈ T.
θ∈Ω
Then {Φt } has exponential dichotomy on Ω.
Proof of Lemma 4.10. The assumptions imply σ(EΨ ) ∩ T = ∅, where
(zI − EΨ )−1 f (θ) = [zI − Φp(θ) (θ)]−1 f (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 Mañe sequence, the cocycle {Φt } does not have exponential dichotomy on Ω due to Proposition 4.2. Therefore the assumptions in Lemma 4.10 cannot
be satisfied. This means that either
(1) kzI − Φp(θn ) (θn )k•,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
kzf − E p(θ0 ) f − gk∞ ≥ |zf (θ0 ) − Φp(θ0 ) f (θ0 ) − y| ≥ δ;
that is, z ∈ σr (E p(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
Corollary 4.11. Let {Φt }t∈R+ be an exponentially bounded, strongly continuous cocycle
{Φt }t∈R+ over a continuous flow {ϕt }t∈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 Mañe 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 Mañe sequence {θm }∞
1 .
Proof. If σap (E) ∩ T 6= ∅, then a Maǹe sequence exists by Proposition 4.4. If a Mañe
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 Mañe 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 (Φp(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 {E t } 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 kzI − Φp(θn ) (θn )k•,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
{ϕ̂t } on Θ × X to the existence and uniqueness of the solution u for the mild form of
the inhomogeneous variational equation u̇(ϕt θ) = (A(ϕt θ) − λ)u(ϕt θ) + g(ϕt θ). Recall
that Φtλ (θ) = e−λt Φt (θ) for λ ∈ C, t ≥ 0, and θ ∈ Θ. Consider the mild inhomogeneous
equation
Z
t
τ
τ
Φt−τ
λ (ϕ θ)g(ϕ θ) dτ,
u(ϕt θ) = Φtλ (θ)u(θ) +
t ≥ 0, θ ∈ Θ,
(5.1)
0
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
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 := Rλ gn → u in F. Since (5.1) holds for un
and gn , we have that u = Rλ g:
u(θ) = lim un (θ) = lim
n→∞
n→∞
Φtλ (ϕ−t θ)un (ϕ−t θ)
= Φtλ (ϕ−t θ)u(ϕ−t θ) +
Z
+
t
τ −t
Φt−τ
θ)gn (ϕτ −t θ) dτ
λ (ϕ
0
Z
t
τ −t
Φt−τ
θ)g(ϕτ −t θ) dτ.
λ (ϕ
0
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 {Φt }t∈R+ be a strongly continuous cocycle {Φt }t∈R+ over a continuous flow {ϕt }t∈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
t
t ≥ 0, θ ∈ Θ,
ΦP (θ) = Φt (θ)P (θ),
G(θ, t) =
−t
t
t
t
−1
−ΦQ (θ) = −[ΦQ (ϕ θ)Q(ϕ θ)] , t < 0, θ ∈ Θ.
Clearly, kG(θ, t)k ≤ M e−β|t| and (θ, t) 7→ G(θ, t) is strongly continuous on Θ × (R\{0}).
Moreover, Green’s operator Ĝ defined by
Z
∞
(Ĝf )(θ) =
G(ϕ−τ θ, τ )f (ϕ−τ θ) dτ
for f ∈ F and θ ∈ Θ
−∞
is a bounded operator on C0 (Θ; X) and Cb (Θ; X). To see that Ĝ 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 := sup{t ≥ 0 : ϕτ (θn ) ∈
/K
for all τ ∈ [−t, t]}.
Suppose that tn < T < ∞. Then there exists τn ∈ [−T, T ] such that ϕτn θn ∈ K for
S
n ∈ N. This means that θn ∈ |τ |≤T ϕτ (K), a contradiction. As a result, tn → ∞ and
(Ĝf )(θn ) → 0 due to the estimate kG(θ, τ )k ≤ M e−β|τ | .
28
Let u = Ĝg for g ∈ F. We compute
u(ϕt θ) − Φt (θ)u(θ)
Z ∞
Z ∞
τ
t−τ
t−τ
t−τ
t
=
Φ (ϕ θ)P (ϕ θ)g(ϕ θ) dτ − Φ (θ)
Φτ (ϕ−τ θ)P (ϕ−τ θ)g(ϕ−τ θ) dτ
0
0
Z 0
Z 0
τ
t−τ
t−τ
t−τ
t
ΦτQ (ϕ−τ θ)Q(ϕ−τ θ)g(ϕ−τ θ) dτ
ΦQ (ϕ θ)Q(ϕ θ)g(ϕ θ)d τ + Φ (θ)
−
−∞
−∞
Z
t
Φt−τ (ϕτ θ)P (ϕτ θ)g(ϕτ θ)dτ −
=
0
Φt−τ (ϕτ θ)P (ϕτ θ)g(ϕτ θ) dτ
−∞
−∞
Z
Z
∞
τ
τ
τ
Φt−τ
Q (ϕ θ)Q(ϕ θ)g(ϕ θ) dτ
Z
∞
τ
τ
τ
−
Φt−τ
+
Q (ϕ θ)Q(ϕ θ)g(ϕ θ) dτ
t
0
Z t
Z t
Φt−τ (ϕτ θ)Q(ϕτ θ)g(ϕτ θ) dτ
Φt−τ (ϕτ θ)P (ϕτ θ)g(ϕτ θ) dτ +
=
0
0
Z t
Φt−τ (ϕτ θ)g(ϕτ θ)dτ
=
0
for t ≥ 0 and θ ∈ Θ. This proves that u = Ĝg satisfies (5.1) with λ = 0.
To show uniqueness, take g = 0 and let u ∈ F satisfy u(ϕt θ) = Φt (θ)u(θ) for θ ∈ Θ
and t ≥ 0. Since {Φt } has exponential dichotomy, we have
P (θ)u(θ) = ΦtP (ϕ−t θ)P (ϕ−t θ)u(ϕ−t θ) and Q(θ)u(θ) = [ΦtQ (θ)]−1 Q(ϕt θ)u(ϕt θ)
for t ≥ 0 and θ ∈ Θ. Thus, the estimates
|P (θ)u(θ)| ≤ M e−βt kuk∞
and |Q(θ)u(θ)| ≤ M e−βt kuk∞
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 {Φt }t∈R+ be an exponentially bounded, strongly continuous cocycle
over a continuous flow {ϕt }t∈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 exponential 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 = Rλ g ∈ C0 (Θ; X). Applying E h in (5.1), we obtain
Z h
−λh
h
h
−h
−h
τ −h
e (E u)(θ) = Φλ (ϕ θ)u(ϕ θ) = u(θ) −
Φh−τ
θ)g(ϕτ −h θ) dτ.
λ (ϕ
0
Therefore,
Z
1 h τ −τ
− u(θ)] = −
Φ (ϕ θ)g(ϕ−τ θ) dτ,
h 0 λ
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).
1 −λh
[e (E h u)(θ)
h
29
If (Γ − λI)u = 0, then e−λt E t u = u or u(ϕt θ) = 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 ϕ1 θ = 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
R p(θ0 )
β(τ )dτ = 1. On the orbit O(θ0 ) define
0
h
i
p(θ )+τ
τ
τ
1
g(ϕ θ0 ) = β(τ )Φλ (θ0 )y + p(θ0 ) − β(τ ) Φλ 0 (θ0 )y.
The function g is continuous on O(θ0 ). Extend g on Θ continuously to a function g ∈ F
with a compact support. Let u = Rλ g. Observe that
p(θ )−τ
p(θ )+τ
Φλ 0 (ϕτ θ0 )Φλ 0 (θ0 ) = Φ2p(θ0 ) (θ0 ).
R p(θ )
Using this fact, (5.1), and 0 0 [ p(θ10 ) − β(τ )]dτ = 0, one obtains
p(θ0 )
u(ϕp(θ0 ) θ0 ) = u(θ0 ) = Φλ
p(θ0 )
(θ0 )u(θ0 ) + Φλ
(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 {Φt }t∈R+ be an exponentially bounded, strongly continuous cocycle
over a continuous flow {ϕt }t∈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 exponential dichotomy on Θ.
Proof. We will show that I − πθ (E) is invertible in L(c0 (Z; X)) and
k(I − πθ (E))−1 kL(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
inf θ∈Θ kI − πθ (E)k•,c0 (Z;X) > 0.
30
(5.3)
Suppose (5.3) does not hold; that is, suppose that
inf θ∈Θ kI − πθ (E)k•,c0 (Z;X) = 0.
(5.4)
By Proposition 4.4, there exists a Mañe 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 kfn k∞ = 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 R0 Γfn = −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
kzI − Φp(θ) (θ)k•,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{kzI − Φp(θ) (θ)k•,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 kfn k∞ ≥ 1/2 and k(Γ − iξ)fn k∞ → 0 for some ξ ∈ R, see (4.11).
Set gn = Γfn −iξfn and λ = iξ. In the next lemma, we show that fn = −Rλ gn . 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 = −Rλ gn ; that is, for t ≥ 0 and θ ∈ Θ one has
t
fn (ϕ θ) −
Φtiξ (θ)fn (θ)
Z
=−
t
t
τ
Φt−τ
iξ (ϕ θ)gn (ϕ θ) dτ.
(5.6)
0
Proof of Lemma 5.8. Recall that fn and gn are supported in Un = {ϕs σ : 0 ≤ s <
p(σ), σ ∈ Σ}, see (4.10) and (4.11). Fix θ = ϕs σ ∈ Un and t ≥ 0.
First, assume that s + t < p(σ). Then, by (4.10),
fn (ϕt θ) − Φtiξ (θ)fn (θ) = fn (ϕt+s σ) − Φtiξ (ϕs σ)fn (ϕs σ)
h
i
p(σ)
p(σ)
t+s
= β(σ)Φt+s
(σ)
α(
)
(x
−
Φ
x
)
+
Φ
(σ)x
n
n
n
iξ
iξ
iξ
p(σ)
h
i
p(σ)
p(σ)
t+s
s
− β(σ)Φiξ (σ) α( p(σ) ) (xn − Φiξ xn ) + Φiξ (σ)xn
h
i
p(σ)
t+s
s
= β(σ)Φt+s
(σ)
α(
)
−
α(
)
(xn − Φiξ xn ).
iξ
p(σ)
p(σ)
31
On the other hand, (4.11) yields
Z t
t−τ
−
Φiξ
(ϕτ θ)gn (ϕτ θ) dτ
0
Z t
1 0 τ +s
p(σ)
t−τ
) dτ
Φiξ
(ϕτ +s σ)Φτiξ+s (σ)[xn − Φiξ (σ)xn ]
α(
= β(σ)
p(σ) p(σ)
0
Z t
dτ
p(σ)
t+s
τ +s
α0 ( p(σ)
)
,
= β(σ)Φiξ (σ)(xn − Φiξ xn )
p(σ)
0
and (5.6) holds.
Second, assume that s + t ∈ [kp(σ), (k + 1)p(σ)) for k = 1, 2, . . .. Now, ϕt θ = ϕt+s σ =
ϕt+s−kp(σ) σ, where t + s − kp(σ) ∈ [0, p(σ)). The LHS of (5.6) can be written as
t+s−kp(σ)
β(σ) Φiξ
p(σ)
t+s−(k−1)p(σ)
(σ) α( t+s−kp(σ)
) [xn − Φiξ (σ)xn ] + β(σ)Φiξ
p(σ)
t+s+p(σ)
− β(σ) Φiξ
(σ) xn
p(σ)
s
(σ)xn − β(σ) Φt+s
iξ (σ) α( p(σ) ) [xn − Φiξ (σ)xn ].
We split the integral on the RHS of (5.6) in three integrals and compute
Z p(σ)−s
t−τ
−
Φiξ
(ϕτ θ)gn (ϕτ θ) dτ
0
Z
p(σ)−s
s+τ
τ +s
1
α0 ( p(σ)
)Φt−τ
σ)Φτiξ+s (σ)[xn
iξ (ϕ
p(σ)
= β(σ)
p(σ)
− Φiξ (σ)xn ] dτ
0
p(σ)
s
= β(σ) 1 − α( p(σ) ) Φt+s
iξ (σ)[xn − Φiξ (σ)xn ]
since α(1) = 1 by the choice of α. Further,
Z kp(σ)−s
t−τ
Φiξ
(ϕτ θ)gn (ϕτ θ) dτ
−
p(σ)−s
= β(σ)
k−1 Z
X
`=1
(`+1)p(σ)−s
`p(σ)−s
1
p(σ)
s+τ −`p(σ)
−`p(σ)
s+τ −`p(σ)
α0 ( s+τp(σ)
) Φt−τ
σ)Φiξ
iξ (ϕ
(σ)
p(σ)
· [xn − Φiξ (σ)xn ] dτ
t+s−(k−1)p(σ)
= β(σ)[Φiξ
(σ)xn − Φt+s
iξ (σ)xn ],
where we have used α(0) = 0, α(1) = 1, and the cocycle property. Finally,
Z t
τ
τ
Φt−τ
−
iξ (ϕ θ)gn (ϕ θ) dτ
kp(σ)−s
Z
t
= β(σ)
kp(σ)−s
τ +s−kp(σ)
τ +s−kp(σ)
1
α0 ( τ +s−kp(σ)
) Φt−τ
σ)Φiξ
(σ)[xn
iξ (ϕ
p(σ)
p(σ)
t+s−kp(σ)
= β(σ) Φiξ
p(σ)
− Φiξ (σ)xn ] dτ
p(σ)
(σ) α( t+s−kp(σ)
) [xn − Φiξ (σ)xn ].
p(σ)
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
(1) Assume that p(θ0 ) = ∞. For γ ≥ 0, we define the operator
D(γ) : (xk ) 7→ (xk − e−γ(|k|−|k−1|) Φ1 (ϕk−1 θ0 )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
kD(γ)k•,c0 (Z;X) ≥ c for all 0 ≤ γ ≤ γ0 .
(5.8)
Take ȳ = (yk ) ∈ c0 (Z; X)
R with yk = 0 for |k| ≥ n = n(ȳ). Choose a function α ∈ C(R)
with supp α ⊂ (0, 1) and R α(τ )dτ = 1. For m = n + 1, n + 2, · · · and 0 ≤ γ ≤ γ0 , define
g̃m (ϕt θ0 ) = α(t − k + 1) Φt−k+1 (ϕk−1 θ0 ) eγ|k−1| yk−1
for t ∈ [k − 1, k) and k = −m + 1, . . . , m. We can extend g̃m to a function gm ∈ C(Θ; X)
with compact support and
kgm k∞ ≤ kαk∞
sup
kΦτ (θ)k eγ0 n kȳk∞ .
0≤τ ≤1, θ∈Θ
Condition (M0 , Cb (Θ; X)) yields the function um = R0 gm ∈ Cb (Θ; X), that is, one has
Z 1
k
1
k−1
k−1
Φ1−τ (ϕτ +k−1 θ0 )g(ϕτ +k−1 θ0 ) dτ
um (ϕ θ0 ) = Φ (ϕ θ0 )um (ϕ θ0 ) +
0
= Φ1 (ϕk−1 θ0 )um (ϕk−1 θ0 ) + Φ1 (ϕk−1 θ0 )eγ|k−1| yk−1
for k = −m + 1, . . . , m. Set vm = (um (ϕk θ0 ) + 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
krm k∞ ≤ kI − πθ0 (E)kL(`∞ (Z;X)) kum k∞ ≤ kI − πθ0 (E)k kR0 k kgm k∞ ≤ C,
where C depends on ȳ but not on m and γ ∈ [0, γ0 ]. Further, (e−γ|k| vmk )k∈Z ∈ c0 (Z; X)
and
kD(γ)(e−γ|k| vmk )k∈Z − ȳk∞ = k(e−γ|k| [(I − πθ0 (E))vm ]k )k∈Z − ȳk∞
= sup |e−γ|k| rmk | ≤ C e−γm
k∈Z
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 ψ t (θ) = ϕtT (θ) and the
exponentially bounded, strongly continuous cocycle Ψt (θ) = ΦtT (θ) over {ψ t } on Θ. This
cocycle satisfies condition (MλT , Cb (Θ; X)). Also, for the flow {ψ t } one has pψ (θ0 ) = 1. An
application of parts (I) and (II.2.i) to the closed ψ t -invariant set O(θ0 ) and the cocycle
33
Ψ shows that the corresponding operators I − πθ (EΨ ) are invertible on c0 (Z; X) and
k(I − πθ (EΨ ))−1 k ≤ 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
Z t
λt
tA
e x−e x=
e(t−τ )A eλτ y dτ.
0
Consequently, x ∈ D(A) and (λ − A)x = y, i.e., λ ∈
/ σr (A). Lemma 5.5 and (5.3) yield
A
σap (e ) ∩ 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 (eA ) ∩ 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 {Φt }t∈R+ be an exponentially bounded, strongly continuous cocycle
{Φt }t∈R+ over a continuous flow {ϕt }t∈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. Räbiger [private communication] recently gave a direct proof of the following 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 R0 g ∈ 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, Universität Tübingen, Auf der Morgenstelle
10, 72076 Tübingen, Germany.
E-mail address: [email protected]
36