# On Spectral Characterization of Nonuniform Hyperbolicity

Authors: Muna Abu Alhalawa; Davor Dragičević
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We give a complete functional theoretic characterization of tempered exponential dichotomies in terms of the invertibility of certain linear operators acting on a suitable Frechét space. In sharp contrast to previous results, we consider noninvertible linear cocycles acting on infinite-dimensional spaces. The principal advantage of our results is that they avoid the use of Lyapunov norms.


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## Body

## 1. Introduction

The problem of characterizing hyperbolic behaviour of dynamical systems in terms of the spectral properties of certain linear operators has a long history that goes back to the pioneering works of Perron [1] and Li [2]. More precisely, Perron [1] established a complete characterization of the exponential stability of a linear differential equation (1)x′=Atx,in Rn in terms of the solvability (in x) of the nonlinear equation (2)x′=Atx+ft,where f and x belong to suitable function spaces. Similar results for the discrete time dynamics were obtained by Li. The condition that (2) has a (unique) solution x in some space Y1 for any choice of f that belongs to some (possibly different) space Y2 is commonly referred to asadmissibility condition. Clearly, this requirement can be formulated in terms of the linear operator (3)Lxt=x′-Atx,acting between suitable function spaces.The major contribution to this line of the research is due to Massera and Schäffer [3]. Indeed, in contrast to the work of Perron, they have established complete characterization (in terms of admissibility) of the notion of (uniform) exponential dichotomy which includes the notion of exponential stability as a particular case. More precisely, rather than considering only the dynamics that exhibits stable behaviour, they have considered the case of dynamics with the property that the phase space splits into two complementary directions, where in one direction dynamics exhibits stable behaviour while in the complementary direction it possesses an unstable (chaotic) behaviour. In addition, they have developed an axiomatic approach to the problem of constructing all possible pairs (Y1,Y2) of function spaces with the property that the corresponding admissibility condition is equivalent to the existence of exponential dichotomy.To the best of our knowledge, the first results in this direction that deal with the infinite-dimensional dynamics are due to Dalećkiĭ and Kreĭn [4]. For more recent results devoted to continuous and discrete evolution families, we refer to [5–15] for those dealing with uniform exponential behaviour and to [16–20] for those that consider various concepts of nonuniform exponential behaviour.In the context of smooth dynamical systems, first results are due to Mather [21] who proved that a smooth diffeomorphisms f of a compact Riemannian manifold M is Anosov if and only if the operator Γ defined by (4)Γvx=Dff-1xvf-1xon the space of all continuous vector fields v on M is hyperbolic (see [22] for related results in the case of flows). Subsequent related results consider the general case of linear cocycles (or the so-called linear skew product flows) acting on Banach spaces. We refer to [23–29] and references therein. We stress that all of those works consider only uniform hyperbolic behaviour.In the paper [30] devoted to the roughness property of nonuniform hyperbolicity (the so-called tempered dichotomy) for linear cocycles on Banach spaces, the authors posed a questions on whether it is possible to give functional theoretic characterization of nonuniform behaviour. It turns out that the answer to this question is positive and such characterization was developed in [31] (see also [32]) and applied to the above-mentioned roughness property of tempered dichotomies. However, the approach developed in [31] is far from satisfactory since the construction of appropriate spaces on which the Mather type of operator acts is given in terms of the so-called Lyapunov norms which transform nonuniform behaviour into the uniform. Thus, in order to use this characterization to detect nonuniform behaviour, one would first need to construct appropriate Lyapunov norms. Hence, the results in [31] are unfortunately only of limited applicability.In the present paper, we propose an alternative functional theoretic characterization of tempered exponential dichotomies. More precisely, we show that the existence of tempered exponential dichotomy (under the assumptions of the multiplicative ergodic theorem) is equivalent to the invertibility of Mather-type operators acting on a certain Frechét space. Although dealing with Frechét instead of Banach spaces is in principle harder, we feel that nevertheless our results have an advantage over those in [31]. The reason for this is that our Frechét space is built in terms of the original norm and not in terms of Lyapunov norms. We stress that our approach is close in spirit to that developed in [33] for Lyapunov regular trajectories in the finite-dimensional setting, associated with a smooth diffeomorphism.In order to formulate an explicit result, let(Ω,F,P,σ) be an invertible and ergodic measure preserving dynamical system. Moreover, let A be a linear cocycle over this system which takes values in a family of compact and injective operators on some Banach space. We will construct a Frechét space Y as well as family of continuous operators Mω:Y→Y, ω∈Ω, such that the following result (which is a combination of Theorems 13 and 14) is valid.Theorem 1.
The cocycleA admits a tempered dichotomy if and only if Id-Mω is an invertible operator on Y for P-a.e. ω∈Ω, where Id denotes the identity operator on Y.We hope that our results will be applicable to the study of nonuniformly hyperbolic dynamical systems. We emphasize that since the landmark works of Oseledets [34] and in particular Pesin [35] this theory has become one of the central themes of the modern dynamical systems theory (see [36] for a detailed exposition). We refer to [37–39] and references therein for the discussions regarding the various extensions of this theory to the infinite-dimensional setting.


## 2. Preliminaries

In this section we recall some notions and collect previous auxiliary results that will be used in the following section.


### 2.1. Linear Cocycles and Tempered Exponential Dichotomy

Consider a probability space(Ω,F,P) and assume that σ:Ω→Ω is an invertible, P-preserving transformation which is ergodic. Furthermore, let X be a separable Banach space and denote by B(X) the space of all bounded linear operators on X. Finally, let N0={0,1,2,…}.We say that a mapA:Ω×N0→B(X) is alinear cocycle over σ if(1)
A(ω,0)=Id for ω∈Ω;(2)
A(ω,n+m)=A(σm(ω),n)A(ω,m) for ω∈Ω and n,m∈N0;(3)
(5)∫Ωlog+AωdPω<∞,where(6)Aω=Aω,1,forω∈Ω;(4)
ω↦A(ω)x is a measurable map from Ω to X for each x∈X.We recall that the mapA given by (6) is called thegenerator of a cocycle A.We also introduce the notion of a tempered random variable. We say that a measurable mapK:Ω→(0,∞) is atempered random variable if (7)limn→±∞1nlog⁡Kσnω=0,forP-a.e.ω∈Ω.The following well-known result (see [40] for example) will be useful in our arguments.Proposition 2.
Assume thatK:Ω→(0,∞) is a tempered random variable. Then, for ε>0, there exists a measurable map C:Ω→(0,∞) such that (1)
ForP-a.e. ω∈Ω, (8)Kω≤Cω.(2)
ForP-a.e. ω∈Ω and every n∈Z, we have that (9)Cσnω≤Cωeεn.Finally, we recall the notion of a tempered exponential dichotomy [30, 31]. We say that the cocycle A with generator A as in (6) admits atempered exponential dichotomy if there exist λ>0, a tempered random variable K:Ω→(0,∞), and a family of projections P(ω)∈B(X), ω∈Ω such that (1)
ω↦P(ω)x is a measurable map for each x∈X;(2)
forP-a.e. ω∈Ω,(10)AωPω=PσωAω,and the map(11)Aω∣KerPω:KerPω⟶KerPσωis invertible;(3)
forP-a.e. ω∈Ω and every n∈N, (12)Aω,nPω≤Kωe-λn,(13)Aω,-nId-Pω≤Kωe-λn,where(14)Aω,-n≔Aσ-nω,n∣KerPσ-nω-1is a well-defined linear map fromKerP(ω) to KerP(σ-n(ω)).


### 2.2. Multiplicative Ergodic Theorem

We now recall the version of the multiplicative ergodic theorem established by Lian and Lu [39]. For the sake of simplicity, we will restrict our attention to the case of compact cocycles.Theorem 3.
Assume thatA is a linear cocycle such that A(ω) is a compact and injective operator for P-a.e. ω∈Ω, where A(ω) is given by (6). Then either of the following is true. (1)
There is a finite sequence of numbers(15)λ1>λ2>⋯>λk>λ∞=-∞,and a decomposition(16)X=E1ω⊕⋯⊕Ekω⊕E∞ω,such that, forP-a.e. ω∈Ω, (17)AωEiω=Eiσω,i=1,…,k,AωE∞ω⊂E∞σω,limn→∞1nlog⁡Aω,nx=λiforx∈Eiω∖0,i∈1,…,k,limn→∞1nlog⁡Aω,nx=λ∞forx∈E∞ω.Moreover, eachEi(ω), i=1,…,k is a finite-dimensional subspace of X.(2)
There exists an infinite sequence of numbers(18)λ1>λ2>⋯>λk>⋯>λ∞=-∞,and for eachk∈N a decomposition (19)X=E1ω⊕⋯⊕Ekω⊕Fkω,such that, forP-a.e. ω∈Ω, (20)AωEiω=Eiσω,i=1,…,k,AωFkω⊂Fkσω,limn→∞1nlog⁡Aω,nx=λi,forx∈Eiω∖0,i=1,…,k,limsupn→∞1nlog⁡Aω,nx≤λi+1,forx∈Fkω.Moreover, eachEi(ω), i≠∞, is a finite-dimensional subspace of X.We note that the numbersλi are calledLyapunov exponents of the cocycle A. In addition, subspaces Ei(ω) are calledOseledets subspaces. Now we are in position to state sufficient conditions for the existence of tempered exponential dichotomy. The following result was established by Lian and Lu [39].Theorem 4.
Assume thatA is a linear cocycle satisfying assumptions of Theorem 3. If all Lyapunov exponents of A are nonzero, then A admits a tempered exponential dichotomy.


### 2.3. Frechét Space

We now introduce our Frechét space that will play a central role in our arguments. Set(21)Y=x=xnn∈Z⊂X:limsupn→∞1nlog⁡xn≤0.It is easy to verify that Y is a vector space. Furthermore, for each k∈N, set (22)Yk=x=xnn∈Z⊂X:xk<∞,where (23)xk=supn∈Zxne-n/k.It is straightforward to show that (i)
(Yk,·k) is a Banach space for each k∈N;(ii)
(24)Y=⋂k∈NYk;(iii)
for eachk∈N(25)Yk+1⊂Yk;(iv)
xk≤xk+1 for every x∈Y and k∈N.It follows from the above properties that we can equipY with the structure of thegraded Frechét space by saying that the sequence (xl)l∈N⊂Y converges to x∈Y if and only if (xl)l∈N converges to x in Yk for each k∈N.


### 2.4. Mather-Type Operator

The other crucial ingredient in our characterization is the construction of appropriate linear operators. LetA be a linear cocycle and consider its generator A given by (6). For ω∈Ω, we define (26)Mωxn=Aσn-1ωxn-1,forn∈Z,x=xnn∈Z⊂X.Obviously, Mω is a linear map. Let us now establish several auxiliary results.Lemma 5.
ForP-a.e. ω∈Ω, (27)Mω:Yk+1⟶Ykis a well-defined and bounded linear operator for each k∈N.Proof.
It follows from (5) that there exist a>0 and a tempered random variable K:Ω→(0,∞) such that (28)Aω,n≤Kωean,for P-a.e. ω∈Ω and every n∈N. Fix now k∈N and choose ε>0 such that ε<1/k-1/(k+1). Furthermore, let C be given by Proposition 2 and let Ωk′⊂Ω be such that P(Ωk′)=1 and that (8), (9), and (28) hold for each ω∈Ωk′. It follows that (29)e-n/kMωxn=e-n/kAσn-1ωxn-1≤e-n/kKσn-1ωeaxn-1≤e-n/kCσn-1ωeaxn-1≤Cωea-n/k+εn-1xn-1≤Cωea+1/keε-1/kn-1xn-1≤Cωea+1/ke-n-1/k+1xn-1≤Cωea+1/kxk+1,for each n∈Z, x=(xn)n∈Z∈Yk+1, and ω∈Ωk′. We conclude that (30)Mωxk≤Cωea+1/kxk+1,for every x∈Yk+1 and ω∈Ωk′. Set Ω′=⋂k=1∞Ωk′. Then, P(Ω′)=1 and it follows readily from (30) that Mω:Yk+1→Yk is a well-defined and bounded operator for each ω∈Ω′ and k∈N.As a direct consequence of Lemma5, we obtain the following result.Proposition 6.
The operatorMω:Y→Y is a well-defined and continuous operator for P-a.e. ω∈Ω.Proof.
By Lemma5, there exists a full-measure set Ω′⊂Ω such that Mω:Yk+1→Yk is a well-defined and bounded linear operator for each k∈N and ω∈Ω′.
We begin by noting thatMωY⊂Y for each ω∈Ω′. Indeed, for any x∈Y, we have that x∈Yk+1 for each k∈N (see (24)). Hence, Lemma 5 implies that Mωx∈Yk for every k∈N. We conclude that Mωx∈Y.
Let us now establish the continuity ofMω. Take a sequence (xl)l that converges to x in Y. This implies that (xl)l converges to x in Yk+1 for each k∈N. By (30), we have that (Mωxl)l converges to Mωx in Yk for every k∈N and therefore also in Y. The proof is completed.


## 2.1. Linear Cocycles and Tempered Exponential Dichotomy

Consider a probability space(Ω,F,P) and assume that σ:Ω→Ω is an invertible, P-preserving transformation which is ergodic. Furthermore, let X be a separable Banach space and denote by B(X) the space of all bounded linear operators on X. Finally, let N0={0,1,2,…}.We say that a mapA:Ω×N0→B(X) is alinear cocycle over σ if(1)
A(ω,0)=Id for ω∈Ω;(2)
A(ω,n+m)=A(σm(ω),n)A(ω,m) for ω∈Ω and n,m∈N0;(3)
(5)∫Ωlog+AωdPω<∞,where(6)Aω=Aω,1,forω∈Ω;(4)
ω↦A(ω)x is a measurable map from Ω to X for each x∈X.We recall that the mapA given by (6) is called thegenerator of a cocycle A.We also introduce the notion of a tempered random variable. We say that a measurable mapK:Ω→(0,∞) is atempered random variable if (7)limn→±∞1nlog⁡Kσnω=0,forP-a.e.ω∈Ω.The following well-known result (see [40] for example) will be useful in our arguments.Proposition 2.
Assume thatK:Ω→(0,∞) is a tempered random variable. Then, for ε>0, there exists a measurable map C:Ω→(0,∞) such that (1)
ForP-a.e. ω∈Ω, (8)Kω≤Cω.(2)
ForP-a.e. ω∈Ω and every n∈Z, we have that (9)Cσnω≤Cωeεn.Finally, we recall the notion of a tempered exponential dichotomy [30, 31]. We say that the cocycle A with generator A as in (6) admits atempered exponential dichotomy if there exist λ>0, a tempered random variable K:Ω→(0,∞), and a family of projections P(ω)∈B(X), ω∈Ω such that (1)
ω↦P(ω)x is a measurable map for each x∈X;(2)
forP-a.e. ω∈Ω,(10)AωPω=PσωAω,and the map(11)Aω∣KerPω:KerPω⟶KerPσωis invertible;(3)
forP-a.e. ω∈Ω and every n∈N, (12)Aω,nPω≤Kωe-λn,(13)Aω,-nId-Pω≤Kωe-λn,where(14)Aω,-n≔Aσ-nω,n∣KerPσ-nω-1is a well-defined linear map fromKerP(ω) to KerP(σ-n(ω)).


## 2.2. Multiplicative Ergodic Theorem

We now recall the version of the multiplicative ergodic theorem established by Lian and Lu [39]. For the sake of simplicity, we will restrict our attention to the case of compact cocycles.Theorem 3.
Assume thatA is a linear cocycle such that A(ω) is a compact and injective operator for P-a.e. ω∈Ω, where A(ω) is given by (6). Then either of the following is true. (1)
There is a finite sequence of numbers(15)λ1>λ2>⋯>λk>λ∞=-∞,and a decomposition(16)X=E1ω⊕⋯⊕Ekω⊕E∞ω,such that, forP-a.e. ω∈Ω, (17)AωEiω=Eiσω,i=1,…,k,AωE∞ω⊂E∞σω,limn→∞1nlog⁡Aω,nx=λiforx∈Eiω∖0,i∈1,…,k,limn→∞1nlog⁡Aω,nx=λ∞forx∈E∞ω.Moreover, eachEi(ω), i=1,…,k is a finite-dimensional subspace of X.(2)
There exists an infinite sequence of numbers(18)λ1>λ2>⋯>λk>⋯>λ∞=-∞,and for eachk∈N a decomposition (19)X=E1ω⊕⋯⊕Ekω⊕Fkω,such that, forP-a.e. ω∈Ω, (20)AωEiω=Eiσω,i=1,…,k,AωFkω⊂Fkσω,limn→∞1nlog⁡Aω,nx=λi,forx∈Eiω∖0,i=1,…,k,limsupn→∞1nlog⁡Aω,nx≤λi+1,forx∈Fkω.Moreover, eachEi(ω), i≠∞, is a finite-dimensional subspace of X.We note that the numbersλi are calledLyapunov exponents of the cocycle A. In addition, subspaces Ei(ω) are calledOseledets subspaces. Now we are in position to state sufficient conditions for the existence of tempered exponential dichotomy. The following result was established by Lian and Lu [39].Theorem 4.
Assume thatA is a linear cocycle satisfying assumptions of Theorem 3. If all Lyapunov exponents of A are nonzero, then A admits a tempered exponential dichotomy.


## 2.3. Frechét Space

We now introduce our Frechét space that will play a central role in our arguments. Set(21)Y=x=xnn∈Z⊂X:limsupn→∞1nlog⁡xn≤0.It is easy to verify that Y is a vector space. Furthermore, for each k∈N, set (22)Yk=x=xnn∈Z⊂X:xk<∞,where (23)xk=supn∈Zxne-n/k.It is straightforward to show that (i)
(Yk,·k) is a Banach space for each k∈N;(ii)
(24)Y=⋂k∈NYk;(iii)
for eachk∈N(25)Yk+1⊂Yk;(iv)
xk≤xk+1 for every x∈Y and k∈N.It follows from the above properties that we can equipY with the structure of thegraded Frechét space by saying that the sequence (xl)l∈N⊂Y converges to x∈Y if and only if (xl)l∈N converges to x in Yk for each k∈N.


## 2.4. Mather-Type Operator

The other crucial ingredient in our characterization is the construction of appropriate linear operators. LetA be a linear cocycle and consider its generator A given by (6). For ω∈Ω, we define (26)Mωxn=Aσn-1ωxn-1,forn∈Z,x=xnn∈Z⊂X.Obviously, Mω is a linear map. Let us now establish several auxiliary results.Lemma 5.
ForP-a.e. ω∈Ω, (27)Mω:Yk+1⟶Ykis a well-defined and bounded linear operator for each k∈N.Proof.
It follows from (5) that there exist a>0 and a tempered random variable K:Ω→(0,∞) such that (28)Aω,n≤Kωean,for P-a.e. ω∈Ω and every n∈N. Fix now k∈N and choose ε>0 such that ε<1/k-1/(k+1). Furthermore, let C be given by Proposition 2 and let Ωk′⊂Ω be such that P(Ωk′)=1 and that (8), (9), and (28) hold for each ω∈Ωk′. It follows that (29)e-n/kMωxn=e-n/kAσn-1ωxn-1≤e-n/kKσn-1ωeaxn-1≤e-n/kCσn-1ωeaxn-1≤Cωea-n/k+εn-1xn-1≤Cωea+1/keε-1/kn-1xn-1≤Cωea+1/ke-n-1/k+1xn-1≤Cωea+1/kxk+1,for each n∈Z, x=(xn)n∈Z∈Yk+1, and ω∈Ωk′. We conclude that (30)Mωxk≤Cωea+1/kxk+1,for every x∈Yk+1 and ω∈Ωk′. Set Ω′=⋂k=1∞Ωk′. Then, P(Ω′)=1 and it follows readily from (30) that Mω:Yk+1→Yk is a well-defined and bounded operator for each ω∈Ω′ and k∈N.As a direct consequence of Lemma5, we obtain the following result.Proposition 6.
The operatorMω:Y→Y is a well-defined and continuous operator for P-a.e. ω∈Ω.Proof.
By Lemma5, there exists a full-measure set Ω′⊂Ω such that Mω:Yk+1→Yk is a well-defined and bounded linear operator for each k∈N and ω∈Ω′.
We begin by noting thatMωY⊂Y for each ω∈Ω′. Indeed, for any x∈Y, we have that x∈Yk+1 for each k∈N (see (24)). Hence, Lemma 5 implies that Mωx∈Yk for every k∈N. We conclude that Mωx∈Y.
Let us now establish the continuity ofMω. Take a sequence (xl)l that converges to x in Y. This implies that (xl)l converges to x in Yk+1 for each k∈N. By (30), we have that (Mωxl)l converges to Mωx in Yk for every k∈N and therefore also in Y. The proof is completed.


## 3. Main Results

In this section we obtain the main results of this paper; that is, we establish the complete characterization of tempered exponential dichotomies in terms of the invertibility of operatorsId-Mω on the space Y. Before we establish several auxiliary lemmas, we will introduce some additional notation. Assume that the cocycle A admits a tempered dichotomy and let P(ω), ω∈Ω, be the associated family of projections. For ω∈Ω, we define linear operators Γωi, i=1,2 by (31)Γω1xn=∑m=0∞Aσn-mω,mPσn-mωxn-m,(32)Γω2xn=∑m=1∞Aσn+mω,-mId-Pσn+mωxn+m,where x=(xn)n∈Z.Lemma 7.
Assume that the cocycleA admits a tempered dichotomy. Then, Γω1:Yk+1→Yk given by (31) is a well-defined and bounded linear operator for P-a.e. ω∈Ω and sufficiently large k∈N.Proof.
Take an arbitraryk∈N satisfying 1/k<λ and choose ε∈(0,1/k-1/(k+1)). Furthermore, let C be given by Proposition 2 (with respect to K as in the notion of tempered dichotomy). It follows from (8), (9), and (12) that (33)e-n/kΓω1xn≤e-n/k∑m=0∞Kσn-mωe-λmxn-m≤e-n/k∑m=0∞Cσn-mωe-λmxn-m≤Cωe-n/k∑m=0∞eεn-me-λmxn-m≤Cω∑m=0∞eε-1/kn-me1/k-λmxn-m≤Cω∑m=0∞e-n-m/k+1e1/k-λmxn-m≤Cω1-e1/k-λxk+1,for P-a.e. ω∈Ω and every n∈Z and x=(xn)n∈Z∈Yk+1. Hence, (34)Γω1xk≤Cω1-e1/k-λxk+1,for P-a.e. ω∈Ω and every x=(xn)n∈Z∈Yk+1. The proof of the lemma is completed.The following result follows from Lemma7 in the same way Proposition 6 follows from Lemma 5.Proposition 8.
Assume that the cocycleA admits a tempered dichotomy. Then, the operator Γω1:Y→Y given by (31) is a well-defined and continuous operator for P-a.e. ω∈Ω.Proof.
By Lemma7, there exists a full-measure set Ω′⊂Ω such that Γω1:Yk+1→Yk is a well-defined and bounded linear operator for each k∈N sufficiently large and ω∈Ω′.
We begin by noting thatΓω1Y⊂Y for each ω∈Ω′. Indeed, for any x∈Y, we have that x∈Yk+1 for each k∈N (see (24)). Hence, Lemma 7 implies that Γω1x∈Yk for every k∈N sufficiently large. Hence, (24) and (25) imply that Γω1x∈Y.
We now prove thatΓω1 is continuous. Take a sequence (xl)l that converges to x in Y. This implies that (xl)l converges to x in Yk+1 for each k∈N. By (34), we have that (Γω1xl)l converges to Γω1x in Yk for every k∈N sufficiently large. This implies that (Γω1xl)l converges to Γω1x in Y.Lemma 9.
Assume that the cocycleA admits a tempered dichotomy. Then, the operator Γω2:Yk+1→Yk given by (32) is well-defined and bounded for P-a.e. ω∈Ω and sufficiently large k∈N.Proof.
Using the same notation as in the proof of Lemma7, it follows from (8), (9), and (12) that (35)e-n/kΓω2xn≤e-n/k∑m=1∞Kσn+mωe-λmxn+m≤e-n/k∑m=1∞Cσn+mωe-λmxn+m≤Cωe-n/k∑m=1∞eεn+me-λmxn+m≤Cω∑m=1∞e-n+m/k+1e1/k-λmxn+m≤Cωe1/k-λ1-e1/k-λxk+1,for μ-a.e. ω∈Ω and every n∈Z and x=(xn)n∈Z∈Yk+1. Hence, (36)Γω2xk≤Cωe1/k-λ1-e1/k-λxk+1,for P-a.e. ω∈Ω and every x=(xn)n∈Z∈Yk+1. This immediately yields the conclusion of the lemma.Lemma9 implies the following result whose proof is analogous to the proof of Proposition 8.Proposition 10.
Assume that the cocycleA admits a tempered dichotomy. Then, the operator Γω2:Y→Y given by (32) is a well-defined and continuous operator for P-a.e. ω∈Ω.Proposition 11.
Assume that the cocycleA admits a tempered dichotomy. Then, one has the following: (1)
the operatorΓω≔Γω1-Γω2:Yk+1→Yk is well-defined and bounded for P-a.e. ω∈Ω and sufficiently large k∈N, where Γω1 and Γω2 are given by (31) and (32), respectively;(2)
the operatorΓω:Y→Y is a well-defined and continuous operator for P-a.e. ω∈Ω.Proof.
The first assertion follows from Lemmas7 and 9, while the second follows from Propositions 8 and 10.The connection betweenΓω and Mω is given by the following result.Lemma 12.
Assume that the cocycleA admits a tempered dichotomy. Then, (37)ΓωId-Mωx=x,for P-a.e. ω∈Ω, k sufficiently large, and every x∈Yk+1. In particular, (38)ΓωId-Mωx=xforx∈Y.Proof.
Observe that (recall thatA is given by (6)) (39)Γω1Id-Mωxn=∑m=0∞Aσn-mω,mPσn-mωxn-m-∑m=0∞Aσn-mω,mPσn-mωAσn-m-1ωxn-m-1=∑m=0∞Aσn-mω,mPσn-mωxn-m-∑m=0∞Aσn-m-1ω,m+1Pσn-m-1ωxn-m-1=Pσnωxn.Similarly, we have that (40)Γω2Id-Mωxn=∑m=1∞Aσn+mω,-mId-Pσn+mωxn+m-∑m=1∞Aσn+mω,-mId-Pσn+mωAσn+m-1ωxn+m-1=∑m=1∞Aσn+mω,-mId-Pσn+mωxn+m-∑m=1∞Aσn+m-1ω,-m-1Id-Pσn+m-1ωxn+m-1=-Id-Pσnωxn.Hence, (41)ΓωId-Mωxn=Γω1Id-Mωxn-Γω2Id-Mωxn=Pσnωxn+Id-Pσnωxn=xn,which yields the first statement of the lemma. The second statement is a direct consequence of the first.Theorem 13.
Assume that the cocycleA admits a tempered dichotomy. Then, Id-Mω is an invertible continuous operator on Y for P-a.e. ω∈Ω.Proof.
By Lemma12, we have that (42)ΓωId-Mω=IdonY,for P-a.e. ω∈Ω. Let us now prove that (43)Id-MωΓω=IdonY,for P-a.e. ω∈Ω. Observe that (A is again as in (6)) (44)Γω1x-MωΓω1xn=∑m=0∞Aσn-mω,mPσn-mωxn-m-Aσn-1ω∑m=0∞Aσn-m-1ω,mPσn-m-1ωxn-m-1=∑m=0∞Aσn-mω,mPσn-mωxn-m-∑m=0∞Aσn-m-1ω,m+1Pσn-m-1ωxn-m-1=Pσnωxn,and thus (45)Γω1x-MωΓω1xn=Pσnωxn,for P-a.e. ω∈Ω, every n∈Z, and x=(xn)n∈Z∈Y. Similarly, (46)-Γω2x+MωΓω2xn=-∑m=1∞Aσn+mω,-mId-Pσn+mωxn+m+Aσn-1ω∑m=1∞Aσn+m-1ω,-mId-Pσn+m-1ωxn+m-1=-∑m=1∞Aσn+mω,-mId-Pσn+mωxn+m+∑m=1∞Aσn+m-1ω,-m-1Id-Pσn+m-1ωxn+m-1=Id-Pσnωxn,and therefore (47)-Γω2x+MωΓω2xn=Id-Pσnωxn,for P-a.e. ω∈Ω, every n∈Z, and x=(xn)n∈Z∈Y. Finally, we observe that (45) and (47) readily imply (43).We now establish the converse of Theorem13.Theorem 14.
LetA be a linear cocycle with generator A as in (6) such that A(ω) is injective and compact operator for P-a.e. ω∈Ω. Furthermore, assume that Id-Mω is an invertible operator on Y for P-a.e. ω∈Ω. Then, the cocycle A admits a tempered dichotomy.Proof.
In a view of Theorem4, it is sufficient to show that all Lyapunov exponents of A are nonzero. Assume the opposite, that is, zero is a Lyapunov exponent of A and let E0(ω) denote the corresponding Oseledets subspace. Then, Theorem 3 implies that (48)limn→±∞1nlog⁡Aω,nv=0,forP-a.e.ω∈Ω,0≠v∈E0ω.Fix 0≠v∈E0(ω) and consider a sequence x=(xn)n∈Z⊂X defined by (49)xn=Aω,nv,n∈Z.By (48), x∈Y. It is easy to verify that (Id-Mω)x=x. Since x≠0, we obtain the contradiction with the invertibility of Id-Mω.We note that Theorem1 follows directly from Theorems 13 and 14.Remark 15.
We remark that Theorems3 and 4 are in fact valid under weaker assumption that the cocycle A is quasicompact and that A(ω) given by (6) is injective for P-a.e. ω∈Ω. Consequently, all the results of this paper are also valid in this setting. We have decided to present our results for the particular case of compact cocycles in order to focus on the novelties of the present paper and to avoid introducing background on quasicompact cocycles (which can be found in [39, 41, 42]). We note that the notion of a quasicompact cocycle is not hard to introduce but it is quite challenging to verify in practice that the cocycle possesses this property.
Furthermore, in principle, the assumption on the injectivity of operatorsA(ω) could be eliminated. Indeed, the most recent versions of the multiplicative ergodic theorem (see [41, 42]) require that the cocycle A is quasicompact and there are no requirements on the injectivity of operators A(ω). Regarding Theorem 4, it is known that in the finite-dimensional setting it holds without injectivity assumptions (see [43, Theorem 2]). We believe that those ideas, when combined with the tools developed in [38], could be extended to the general case of quasicompact cocycles on Banach spaces. However, we refrain from doing so since it would require many technical arguments that would completely overshadow main results of our paper.



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