Strong and weak admissibility of L ∞ spaces

For a dynamics with continuous time, we consider the notion of a strong exponential dichotomy with respect to a family of norms and we characterize it in terms of the admissibility of bounded solutions. Moreover, we consider both strong and weak admissibility, in the sense that the solutions are respectively of a nonautonomous linear equation defined by a strongly continuous function or of an integral equation obtained from perturbing a general evolution family. As a nontrivial application, we establish the robustness of the notions of a strong exponential dichotomy and of a strong nonuniform exponential dichotomy. We emphasize that the last notion is ubiquitous in the context of ergodic theory: for almost all trajectories with nonzero Lyapunov exponents of a measure-preserving flow, the linear variational equation admits a strong nonuniform exponential dichotomy..


Introduction
For a nonautonomous linear equation in a Banach space defined by a strongly continuous function A(t) and more generally for an evolution family T(t, s) in a Banach space, we introduce the notion of a strong exponential dichotomy with respect to a family of norms.This means that besides having the usual upper bounds in the stable direction for positive time and in the unstable direction for negative time, we have, in addition, lower bounds in the stable direction for positive time and in the unstable direction for negative time.Moreover, at each time we consider a possibly different norm.The main motivation comes from ergodic theory.Indeed, for almost all trajectories with nonzero Lyapunov exponents of a measure-preserving flow, the linear variational equation admits a strong nonuniform exponential dichotomy (we refer to [2] for details and references).This last notion is a particular case of the notion of a strong exponential dichotomy with respect to a family of norms, more precisely a family of Lyapunov norms.Therefore, the type of exponential behavior considered in the paper, besides being very common in the context of ergodic theory, plays a unifying role.In particular, it includes as particular cases both the notions of uniform and nonuniform exponential behavior, considering respectively families of constant norms and Lyapunov norms.
Our main aim is to characterize the notion of a strong exponential dichotomy in terms of the admissibility of bounded solutions.The latter corresponds to assume that there exists a unique bounded solution for each time-dependent bounded perturbation of the original dynamics.In addition to considering a nonautonomous linear equation and more generally an arbitrary evolution family, we also consider both strong and weak admissibility, which corresponds to the perturbations of each of those dynamics.More precisely, in the case of equation (1.1) we consider the perturbed equation and its classical solutions, while in the case of an arbitrary evolution family T(t, s) we consider the perturbed integral equation T(t, s)y(s) ds (1.3) and its mild solutions.We refer to the admissibility in the two perturbed equations, respectively, as strong and weak admissibility.We emphasize that a priori none of them implies the other.
Our main results show that: 1. the evolution family defined by equation (1.1) admits a strong exponential dichotomy with respect to a family of norms if and only if it has bounded growth and there exists a unique bounded solution of equation (1.2) for each bounded perturbation y of the original dynamics (see Theorems 2.1 and 2.3); 2. an arbitrary evolution family T(t, s) admits a strong exponential dichotomy with respect to a family of norms if and only if it has bounded growth and there exists a unique bounded solution of equation (1.3) for each bounded perturbation y of the original dynamics (see Theorems 4.1 and 4.2).
Here, "bounded growth" and "bounded" are always with respect to the family of norms • t under consideration.For example, a function y : R → X with values in a Banach space X is said to be bounded (with respect to the norms • t ) if sup t∈R y(t) t < +∞.
For an evolution family with bounded growth defined by a differential equation as in (1.1), it follows from the latter results that there exists a unique bounded solution of equation (1.2) for each bounded perturbation y if and only if there exists a unique bounded solution of equation (1.3) for each bounded perturbation y.In other words, in our setting the notions of weak admissibility and strong admissibility are in fact equivalent.In fact, this can be considered the main contribution of our work.
The study of the admissibility property goes back to pioneering work of Perron in [8] who used it to deduce the stability or the conditional stability under sufficiently small perturbations of a linear equation.For some of the most relevant early contributions in the area we refer to the books by Massera and Schäffer [6] and by Dalec kiȋ and Kreȋn [4].We also refer to [5] for some early results in infinite-dimensional spaces.
As a nontrivial application of these results, we establish the robustness of the notion of a strong exponential dichotomy with respect to a family of norms and of a strong nonuniform exponential dichotomy.This corresponds to show that any sufficiently small linear perturbation of the dynamics is still, respectively, a strong exponential dichotomy with respect to a family of norms and a strong nonuniform exponential dichotomy.We emphasize that the study of robustness has a long history; see in particular [3,7,9,10] and the references therein.See also [1] for the study of robustness in the general setting of a nonuniform exponential behavior.

Exponential behavior and strong admissibility 2.1 Exponential dichotomies
Let X = (X, • ) be a Banach space and let B(X) be the set of all bounded linear operators on X.A function A : R → B(X) is said to be strongly continuous if for each x ∈ X the map t → A(t)x is continuous.We note that every continuous function A : R → B(X) is strongly continuous.
Let A : R → B(X) be a strongly continuous function and consider the linear equation Let also T(t, τ) be the associated evolution family.Moreover, we consider a family of norms • t on X for t ∈ R such that: (i) there exist constants C and ε ≥ 0 such that for x ∈ X and t ∈ R; (ii) the map t → x t is measurable for each x ∈ X.
We say that equation (2.1) admits a strong exponential dichotomy with respect to the family of norms • t if: (iii) there exist projections P(t) for t ∈ R such that for t ≥ τ and for t ≤ τ, where Q(τ) = Id − P(τ).

From exponential behavior to admissibility
Let Y be the set of all continuous functions x : R → X such that x ∞ := sup t∈R x(t) t < +∞.
One can easily verify that when equipped with the norm for t ∈ R and thus, x(t) is well defined.Moreover, given t 0 ∈ R, we have and letting τ → +∞ yields that x s (t) = 0 for t ∈ R. Similarly, since x u (t) = T(t, t + τ)x u (t + τ) for τ ≥ 0, we have and hence, x u (t) = 0 for t ∈ R. Therefore, x(t) = 0 for t ∈ R.
It remains to establish the second statement in the theorem.It follows from (2.4) and (2.5) that for t ≤ τ.Therefore, (2.7) holds with K = 2D and a = max{b, −a}.

From admissibility to exponential behavior
Now we establish the converse of Theorem 2.1, that is, we show that if the pair (Y, Y) is admissible, then equation (2.1) admits a strong exponential dichotomy.
Theorem 2.3.Assume that for each y ∈ Y there exists a unique x ∈ Y such that: 1. identity (2.6) holds for t ∈ R; 2. there exist K, a > 0 such that (2.7) holds for x ∈ X and t, τ ∈ R.
Then equation (2.1) admits a strong exponential dichotomy with respect to the family of norms • t .
Proof.Let H be the linear operator defined by Similarly, Since the function s → A(s)x is continuous for each x ∈ X, we have and it follows from the Banach-Steinhaus theorem that M < +∞.Since Hx k → y in Y, we obtain Therefore, which implies that Hx = y and x ∈ D(H).
It follows from Lemma 2.4 and the closed graph theorem that the operator H has a bounded inverse G : Y → Y.
For τ ∈ R, let F s τ be the set of all x ∈ X such that there exists a solution u of equation (2.1) Similarly, let F u τ be the set of all x ∈ X such that there exists a solution u of equation (2.1) with One can easily verify that F s τ and F u τ are subspaces of X. Lemma 2.5.
and thus w(τ) ∈ F s τ .On the other hand, w − u is also a solution of equation (2.1) and sup (w τ and let u be the solution of equation (2.1) with u(τ) = x.It follows from (2.13) and (2.14) that u ∈ Y. Since H is invertible, we must have u = 0 and hence x = 0. Now let P(τ) : X → F s τ and Q(τ) : X → F u τ be the projections associated to the decomposition in (2.15), with P(τ) + Q(τ) = Id.It follows readily from the definitions that property (2.3) holds.
Lemma 2.6.There exists M > 0 such that for x ∈ X and τ ∈ R.
Proof of the lemma.Using the same notation as in the proof of Lemma 2.5, we have where L = sup t∈R |φ (t)|.We note that the constant L is independent of τ.Using (2.7) we obtain g ∞ ≤ LKe a x τ and it follows from (2.17) that This shows that (2.16) holds taking M = 1 + G LKe a .
Proof of the lemma.Let ψ : R → R be a smooth function supported on [τ, +∞) using (2.7) in the last inequality.Hence, using again (2.7), we obtain where C = 2Ke a max{1, G }. Now we show that there exists N ∈ N such that for every τ ∈ R and x ∈ F s τ , Hence, it follows from (2.21) that Letting ε → 0 yields the inequality Hence, property (2.20) holds taking N > 2C 2 G .In order to complete the proof, take t ≥ τ and write t − τ = kN + r, with k ∈ N and 0 ≤ r < N. By (2.16), (2.19) and (2.20), we obtain for x ∈ X. Taking D = 2CM and λ = log 2/K yields inequality (2.18).Lemma 2.8.There exist constants λ, D > 0 such that for x ∈ X and t ≤ τ.
Proof of the lemma.Let ψ : R → R be a smooth function supported on using (2.7) in the last inequality.Hence, using again (2.7), we obtain where C = 2Ke a max{1, G }.
We also show that there exists N ∈ N such that for every τ ∈ R and x ∈ F u τ , x τ for τ − t ≥ N. (2.24) In order to prove (2.24), take t 0 ∈ R such that t 0 < τ and u(t Hence, it follows from (2.25) that Letting ε → 0 yields the inequality Hence, property (2.24) holds taking N > 2C 2 G .Finally, take t ≤ τ and write τ − t = kN + r, with k ∈ N and 0 ≤ r < N. By (2.16), (2.23) and (2.24), we obtain In order to complete the proof of the theorem, we note that it follows from (2.18) and (2.22) that (2.4) holds taking a = −λ and b = λ.Moreover, it follows from (2.7) and (2.16) that (2.5) holds taking D = K(1 + M), a = −a and b = a.

Strong robustness
In this section we establish the robustness of the notion of a strong exponential dichotomy using its characterization in terms of admissibility of the pair (Y, Y) in Theorems 2.1 and 2.3.Theorem 3.1.Let A, B : R → B(X) be strongly continuous functions such that: 1. equation (2.1) admits a strong exponential dichotomy with respect to a family of norms • t satisfying (2.2) for some C > 0 and ε ≥ 0; 2. there exists c ≥ 0 such that If c is sufficiently small, then the equation x = B(t)x admits a strong exponential dichotomy with respect to the same family of norms.
Proof.Let H be the linear operator defined by (2.12) on the domain D(H).For x ∈ D(H) we consider the graph norm Clearly, the operator

Exponential behavior and weak admissibility
In this section we consider a weak form of the admissibility property and we use it to give a characterization of the notion of a strong exponential dichotomy.
A family T(t, τ), for t, τ ∈ R, of bounded linear operators on X is said to be an evolution family if: 3. given t, τ ∈ R and x ∈ X, the maps s → T(t, s)x and s → T(s, τ)x are continuous.
We continue to consider a family of norms • t satisfying conditions (i) and (ii).We say that an evolution family T(t, s) admits a strong exponential dichotomy with respect to the family of norms • t if conditions (iii) and (iv) hold.
We first show that the existence of a strong exponential dichotomy yields the weak admissibility of the pair (Y, Y).Theorem 4.1.If the evolution family T(t, τ) admits a strong exponential dichotomy with respect to the family of norms • t , then: 1. for each y ∈ Y there exists a unique x ∈ Y such that Proof of the lemma.Let (x n ) n∈N be a sequence in D(H) converging to x ∈ Y such that Hx n converges to y ∈ Y.For each τ ∈ R, we have We define a function u : R → X by u(t) = T(t, τ)x.It follows from the definitions of F s τ and F u τ that u ∈ Y.Moreover, Hu = 0 and u ∈ D(H).Since H is invertible, we obtain u = 0 and hence x = 0. Now let P(τ) : X → F s τ and Q(τ) : X → F u τ be the projections associated to the decomposition in (4.2), with P(τ) + Q(τ) = Id.

Lemma 4.5.
There exists M > 0 such that for x ∈ X and τ ∈ R.
Proof of the lemma.Using the same notation as in Lemma 4.4, we have On the other hand, it follows from (2.7) that g ∞ ≤ CKe a x τ , where This shows that (4.3) holds taking M = CKe a G + 1.

Lemma 4.6.
There exist constants λ, D > 0 such that for x ∈ X and t ≥ τ.
We show that there exists N ∈ N such that for every τ ∈ R and x ∈ F s τ , In order to prove (4.4), take t 0 ∈ R such that t 0 > τ and u(t Now take ε > 0 and let ψ : R → R be a smooth function supported on [τ, Clearly, y and v belong to Y and one can easily verify that Hv = y.Therefore, Hence, it follows from (4.7) that Letting ε → 0 yields the inequality Hence, property (4.6) holds taking N > 2C 2 G .Now take t ≥ τ and write t − τ = kN + r, with k ∈ N and 0 ≤ r < N. By (4.3), (4.5) and (4.6), we obtain for x ∈ X. Taking D = 2CM and λ = log 2/K yields property (4.4).

Lemma 4.7.
There exist constants λ, D > 0 such that for x ∈ X and t ≤ τ.

Weak robustness
In a similar manner to that in Section 3 we establish, once more, the robustness of the notion of a strong exponential dichotomy but now using its characterization in terms of the weak admissibility of the pair (Y, Y) in Theorems 4.1 and 4.2.
Theorem 5.1.Assume that the evolution family T(t, τ) admits a strong exponential dichotomy with respect to the family of norms • t and that B : R → B(X) is a strongly continuous function such that B(t) ≤ ce −ε|t| , t ∈ R.
( for t ∈ R and hence, P is a well defined bounded linear operator.Furthermore, it follows from (5.2) that D(H) = D(L) and that H = L + P. For x ∈ D(H) we consider the graph norm Clearly, the operator is bounded and for simplicity we denote it simply by H.Moreover, since H is closed, (D(H), • ∞ ) is a Banach space.By (5.3) we have

Strong nonuniform exponential dichotomies
In this section we consider briefly the notion of a strong nonuniform exponential dichotomy and we obtain a corresponding robustness result.We say that an evolution family T(t, τ), for t, τ ∈ R, admits a strong nonuniform exponential dichotomy if there exists: We first relate this notion to the notion of a strong exponential dichotomy with respect to a family of norms.Proposition 6.1.The following properties are equivalent: 1. T(t, τ) admits a strong nonuniform exponential dichotomy; 2. T(t, τ) admits a strong exponential dichotomy with respect to a family of norms • t satisfying conditions (i) and (ii).
It remains to show that the map t → x t is measurable for each x.Let g(τ) = sup t≥τ T(t, τ)y e −λ(t−τ) .
The following robustness result for the notion of a strong nonuniform exponential dichotomy is an immediate consequence of Theorem 5.1 and Proposition 6.1.Theorem 6.2.Assume that the evolution family T(t, τ) admits a strong nonuniform exponential dichotomy and that B : R → B(X) is a strongly continuous function such that B(t) ≤ ce −ε|t| , t ∈ R.
If c is sufficiently small, then the evolution family U(t, τ) defined by U(t, τ)x = T(t, τ)x + t τ T(t, s)B(s)U(s, τ)x ds, t, τ ∈ R admits a strong nonuniform exponential dichotomy.
Now take ε > 0 and let ψ : R → R be a smooth function supported on [τ, t 0 ] such that 0 ≤ ψ ≤ 1 and ψ = 1 on [τ + ε, t 0 − ε].Moreover, let .20)In order to prove (2.20), take t 0 ∈ R such that t 0 > τ and u(t0 ) t 0 > x τ /2.It follows from (2.19) that 1 2C x τ < u(s) s ≤ C x τ , τ ≤ s ≤ t 0 .(2.21)y(t) = ψ(t)u(t) and v(t) = u(t) t −∞ ψ(s) ds for t ∈ R.Clearly, y and v belong to Y and one can easily verify that Hv = y.Therefore, and sup t∈R |ψ (t)| ≤ 2.Moreover, given x ∈ F u τ , let u be the solution of equation (2.1) with u(τ) = x.It follows from (2.14) that ψu ∈ Y and one can easily verify that H .25) Now take ε > 0 and let ψ : R → R be a smooth function supported on [t 0 , τ] such that 0 ≤ ψ ≤ 1 and ψ = 1 on [t 0 + ε, τ − ε].Moreover, let y(t) = −ψ(t)u(t) and v(t) = u(t) +∞ t ψ(s) ds for t ∈ R.Clearly, y and v belong to Y and one can easily verify that Hv = y.Therefore, One can argue in a similar manner for t ≤ τ.Since L is invertible, it follows from Theorem 2.3 together with Lemma 3.2 that the equation x = B(t)x admits a strong exponential dichotomy with respect to the family of norms • t .
For simplicity, we denote it from now on simply by H.It follows from Lemma 2.4 that (D(H), • ∞ ) is a Banach space.It follows from (2.7) and (3.1) that (B(t) − A(t))x t ≤ cC x t (3.2) for x ∈ X and t ∈ R. We define a linear operator L : D(H) → Y by (Lx)(t) = x (t) − B(t)x(t), t ∈ R. By (3.2) we have (H − L)x ∞ ≤ cC x ∞ (3.3) for x ∈ D(T).By Theorem 2.1, the operator H is invertible.Hence, it follows from (3.3) that if c is sufficiently small, then L is also invertible.Furthermore, it follows from Theorem 2.1 that there exist constants K, a > 0 such that (2.7) holds for x ∈ X and t, τ ∈ R. Now let U(t, τ) be the evolution family associated to the linear equation x = B(t)x.Lemma 3.2.There exist constants K , a > 0 such that U(t, τ)x t ≤ K e a |t−τ| x τ for x ∈ X and t, τ ∈ R. t ≤ Ke a(t−τ) x(τ) τ + K t τ e a(t−s) (B(s) − A(s))x(s) s ds ≤ Ke a(t−τ) x(τ) τ + cCK t τ e a(t−s) x(s) s ds.
Now we establish the converse of Theorem 4.1.Assume that for each y ∈ Y there exists a unique x ∈ Y such that (4.1) holds and that there exist constants K, a > 0 such that (2.7) holds for x ∈ X and t, τ ∈ R. Then the evolution family T(t, τ) admits a strong exponential dichotomy with respect to the family of norms • t .Proof.Let H be the linear operator defined by Hx = y in the domain D(H) formed by all x ∈ Y for which there exists y ∈ Y satisfying (4.1).In order to show that H is well defined, let y 1 , y 2 ∈ Y be such that (s) ds and since the map s → T(t, s)y i (s) is continuous for i = 1, 2, letting τ → t yields that y 1 (t) = y 2 (t) for t ∈ R. The operator H : D(H) → Y is closed.
for t ≥ τ.This shows that property (4.1) holds.It follows readily from (4.1) that the function x is continuous and thus x ∈ Y.The uniqueness of x follows from Lemma 2.2 (that can be obtained using the same proof).This establishes the first property of the theorem.The second property follows exactly as in the proof of Theorem 2.1.
.4) for x ∈ X.On the other hand, by Theorem 4.1, the operator H is invertible.Hence, it follows This shows that property (5.5) holds for t ≥ τ taking K = Kφ(τ) and a = cCK.A similar argument can be used for t ≤ τ.One can now apply Theorem 4.2 to conclude that the evolution family U(t, τ) admits a strong exponential dichotomy.
from(5.4) that if c is sufficiently small, then L is also invertible.It remains to show that there exist K , a > 0 such that U(t, τ)x t ≤ K e a |t−τ| x τ for t, τ ∈ R.