Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

We analyze the existence of no past exponential dichotomies for a well-posed autonomous differential equation that generates a C0-semigroup {T t }t≥0 . The novelty of our approach consists in the fact that we do not assume the T t -invariance of the unstable manifolds. Roughly speaking, we prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space on which the right shift is an isometry for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a no past exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the aforementioned condition is very general since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p spaces, sequence Orlicz spaces, etc. and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.


Introduction
The exponential dichotomy is one of the most basic concepts arising in the theory of dynamical systems.For linear differential equations, the notion was introduced by Perron in 1 , who was concerned with the problem of conditional stability of a system x A t x and its connection with the existence of bounded solutions of the equation x A t x f t, x , where the state space X is a finite-dimensional Banach space and the operator-valued function A • is bounded and continuous in the strong operator topology.Relevant results concerning the extension of Perron's problem in the more general framework of infinitedimensional Banach spaces were obtained by Daleckij and Krein 2 , Bellman 3 , Massera and Schäffer 4 , and more recently by van Neerven 5 , and van Minh et al. 6 .

Sequence Sch äffer Spaces
Let N be the set of all nonnegative integers, N * N \ {0}, R the set of all real numbers, and we will denote by t the greatest integer less than or equal with t.The linear space of all realvalued sequences s : N → R is denoted by S. Let X, • be a real or complex Banach space and consider S X the linear space of all sequences f : N → X.For an X-valued sequence f : N → X, we can associate the sequence f : N → R defining f n f n for all n ∈ N. We also consider two linear operators R, L : S X → S X defined by known as the right shift operator, respectively, the left shift operator.A simple verification gives us LRf f and RLf n f n for n ∈ N * , RLf 0 0, for all f ∈ S X .If A ⊂ N, the characteristic function of A will be denoted by χ A and for the simplicity of notation put δ k χ {k} for each k ∈ N. Definition 2.1.A Banach space E, • E is said to be a sequence Schäffer space if E ⊂ S and the following conditions hold:

2.2
The subspace of ∞ , ∞ 0 {f ∈ ∞ : lim n → ∞ f n 0} often denoted by c 0 with the induced norm is another example of sequence Schäffer space.
It is easy to check that c, • ∞ the space of all convergent sequences is not a sequence Schäffer space.
The spaces 1 , ∞ , and ∞ 0 occupy particularly important positions in the class of sequence Schäffer spaces.For E a sequence Schäffer space, we will define the sequences α E , β E ∈ S by

2.4
For the Banach space Φ , • Φ the conditions s 1 , s 2 , and s 3 are verified; hence Even ∞ is a sequence Orlicz space, obtained from ϕ t 0 for t ∈ 0, 1 and ϕ t ∞ for t > 1 16, Section 3 .
Let x ∈ 0, 1 and m 1/x ∈ N * .Using the fact that Φ −1 is nondecreasing, we have that
Indeed, lim t → 0 Φ t /t 0 and lim t → 0 Φ t /t p ∞, for all p ∈ 1, ∞ , and using the aforementioned remark, our claim follows easilly.Also, Φ / ∞ since χ N ∈ ∞ \ Φ .Remark 2.7.By simple computations we obtain that for 1 ≤ p ≤ ∞ with the convention 1/∞ 0 and for the Orlicz sequence spaces, For two Banach spaces B 1 , For the following three propositions, proofs can be retrieved from 8, Section 3 .
Proposition 2.10.Let E, • E be a sequence Schäffer space.The following characterizations hold: For E, • E being a sequence Schäffer space and X being a Banach space, we consider E X {f ∈ S X : f ∈ E} and f E X f E .To prove that E X , • E X is a Banach space see, for example, 17, Remark 2.1 or 8, Lemma 3.8 .The following properties of this space are simple verifications.
Proposition 2.11.The space E X , • E X is a Banach space with the following properties: To prevent any further confusion, let us fix the notation B X for the class of bounded linear operators acting on X.

Semigroups of Bounded Linear Operators, Exponential Dichotomy, and Admissibility
Let X be a Banach space, x 0 ∈ X, A : D A ⊂ X → X a linear operator, and consider the abstract Cauchy problem A; x 0 : ẋ t Ax t , t ≥ 0, x 0 x 0 .

3.1
The A linear and bounded operator T acting on a complex Banach space is said to be hyperbolic if σ T ∩ Γ ∅ where Γ {z ∈ C : |z| 1} denotes the unit circle in the complex plane and σ T is the spectrum of T .The spectral Riesz projection P for a hyperbolic operator T is given by The projection corresponds to the part of the spectrum of T contained in the open unit disk D 0, 1 .We note that the projection P commutes with T .Since we obtain the spectral radius r T | Im P < 1 and also the operator Hence, if T is hyperbolic, then there exist the constants N, ν > 0 such that, for all integers n ≥ 0, for the previous exposure we consulted 14, page 28 .
Definition 3.2.A semigroup {T t } t≥0 is hyperbolic if there exists t 0 > 0 such that T t 0 is an hyperbolic operator.
Definition 3.3.The semigroup {T t } t≥0 has an exponential dichotomy or that it is exponentially dichotomic if there exist a projection P i.e., P ∈ B X and P 2 P and the constants N, ν > 0 such that i PT t T t P , for all t ≥ 0, ii T t | Ker P : Ker P → Ker P is an isomorphism, for each t ≥ 0, iii T t x ≤ Ne −νt x , for all t ≥ 0 and x ∈ Im P , iv T t −1 | Ker P x ≤ Ne −νt x , for all t ≥ 0 and x ∈ Ker P .
The first condition in the previous definition expresses equivalently that Im P and Ker P are both T t -invariant; the essence of ii is that {T t | Ker P } t≥0 can be extended to a group.The next result establishes the relation between hyperbolicity and exponential dichotomy for C 0 -semigroups on complex Banach spaces.For the proof we refer the reader to 14, Lemma 2.15 , 20, Theorem 1.1 , or alternatively 15 .Proposition 3.4.For a strongly continuous semigroup {T t } t≥0 acting on a complex Banach space the following statements are equivalent.
ii {T t } t≥0 has an exponential dichotomy.
Moreover, if (i) holds, then X X 1 ⊕ X 2 where X 1 Im P , X 2 Im I − P , and is the spectral Riesz projection for T t 0 that corresponds to σ T t 0 ∩ Γ ∅.Also, if {T t } t≥0 has an exponential dichotomy, then σ T t ∩ Γ ∅, for every t ≥ 0.
If A generates an exponentially dichotomic C 0 -semigroup, then the differential equation ẋ Ax has the property that the solutions x • starting from X 1 resp., from X 2 decay exponentially for t > 0 resp., for t < 0 uniformly with respect to the initial data.As it can be seen, the exponential dichotomy concept generalizes strongly the exponential stability concept but it has a serious drawback.It forces the solution that starts from X 2 to exist for negative time, or in counterpart it forces the semigroup to be invertible on X 2 .We will drop off this requirement here and extend the notion of hyperbolicity by replacing the exponential decay in negative time for the solutions starting in X 2 with an exponential blowup in positive time.We will call the "exponential decay on X 1 and exponential blow-up on X 2 " both on positive time behavior as no past exponential dichotomy.Definition 3.5.The semigroup {T t } t≥0 has a no past exponential dichotomy if there exist a projection P and the constants N 1 , N 2 , ν > 0 such that i PT t P T t P , for all t ≥ 0, ii T t x ≤ N 1 e −νt x , for all t ≥ 0, x ∈ Im P , iii T t x ≥ N 2 e νt x , for all t ≥ 0, x ∈ Ker P .Definition 3.6.The semigroup {T t } t≥0 has an ordinary dichotomy if there exist a projection P and the constants N 1 , N 2 > 0 such that i PT t P T t P , for all t ≥ 0, ii T t x ≤ N 1 x , for all t ≥ 0, x ∈ Im P , iii T t x ≥ N 2 x , for all t ≥ 0, x ∈ Ker P .Remark 3.7.Note that if P is one-to-one, then Im P X and thus the concept of no past exponential dichotomy overlaps the concept of exponential stability.Recall that {T t } t≥0 is said to be exponentially stable if one of the following equivalent statements is true: i there exist N, ν > 0 such that T t ≤ Ne −νt for all t ≥ 0; ii there exist t 0 > 0 such that T t 0 < 1.
It is obvious that the existence of an exponential dichotomy implies the existence of a no past exponential dichotomy, but the converse is not valid as the following example points out.
Example 3.8.The following C 0 -semigroup {S t } t≥0 has a no past exponential dichotomy but is not exponentially dichotomic.Let A be a p × p matrix with real entries whose spectrum is contained in the open left-half plane and consider the right shift semigroup on L 1 R , R , given by

3.7
Then S t e tA ⊕ e t T t , S t x, f e tA x, e t T t f has a no past exponential dichotomy on However, the restriction of S t on X 2 is not onto, and thus {S t } t≥0 is not exponentially dichotomic.Lemma 3.9.Let {T t } t≥0 be an exponentially bounded semigroup.If there exist Y , a vector subspace of X, n 0 ∈ N * , H > 0, and η > 1 such that then there exist N, ν > 0 which satisfy T t x ≥ Ne νt x , for all x ∈ Y and t ≥ 0.

3.8
For t ≥ 0, take n t 1 to have Me ω T t x ≥ T n x ≥ He −νn 0 e νn x .Put N : HM −1 e − νn 0 ω > 0 in order to write T t x ≥ Ne νt x .Lemma 3.10.Let {T t } t≥0 be an exponentially bounded semigroup.If there exist Y a vector subspace of X, n 0 ∈ N * , H > 0, and η ∈ 0, 1 such that i T n x ≤ H x , for all n ∈ {0, 1, . . ., n 0 } and x ∈ Y and ii T n n 0 x ≤ η T n x , for all n ∈ N and x ∈ Y , then there exist N, ν > 0 which satisfy T t x ≤ Ne −νt x , for all x ∈ Y and t ≥ 0.
Proof.It is analogous with the proof of Lemma 3.9.

Consider the autonomous inhomogeneous abstract Cauchy problem
x 0 x 0 .

3.9
If A generates the C 0 -semigroup {T t } t≥0 , x ∈ X and f ∈ L 1 loc R , X , then the function u : R → X given by is said to be the mild solution of the Cauchy problem A, f; x .For sufficient conditions assuring that the mild solution is also a classical solution i.e., a continuously differentiable function that verifies the initial value problem A, f; x , we refer the reader to 19, Section 4.2 .
The "test functions method" or "Perron's method" was often used until now see e.g., 1-4 to study properties of asymptotic behavior such as exponential dichotomy.According to Massera and Schäffer 4 by "test functions method" it is meant the relation between certain "test functions" f and "nice solutions" of the inhomogeneous equations A, f : x Ax f.The crudest expression of this method is the notion of admissibility of a pair of classes of functions both in L 1 loc R , X Massera and Schäffer named these classes of functions as the class of "test functions" and the class of "nice solutions" and defined the pair to be admissible if for every "test function" f, the equation A, f has a "nice solution" see 4, Chapter 5, page 124 .
In this spirit, we set the expression of the mild solution of the equation A, f in discrete-time to give the following definition of admissibility in terms of "test sequences" and "nice discrete-time mild solutions".In this way, we do not need any assumption of continuity or measurability and we still obtain continuous-time asymptotic properties for the autonomous differential equation A : x Ax.Definition 3.11.Let E, F be sequence Schäffer spaces.The pair E, F is said to be admissible to {T t } t≥0 if for each f ∈ E X , there exists x ∈ X such that p f •; x ∈ F X , where for each n ∈ N.

Main Results
In this section, for {T t } t≥0 being a semigroup on the Banach space X with M, ω assuring the exponential boundedness if it is the case for {T t } t≥0 to be exponentially bounded and E and F being two sequence Schäffer spaces, we denote which are obviously vector subspaces of X.
Hypothesis 1.The vector subspace X 1,F is closed and admits a closed complement; that is, there exists X 2,F , a closed vector subspace, such that We denote by P 1 the projection onto X 1,F along X 2,F and set P 2 I − P 1 we will prove that in the case of a no past exponential dichotomy for {T t } t≥0 , X 1,F coincides always with X 1 and thus the F-independent notation for projectors is consistent .
Proof.If x ∈ X 1,F and t ≥ 0, then it is to see that T n T t x ≤ T t T n x for all n ∈ N, and since T n x n∈N ∈ F, it follows that T t x ∈ X 1,F .
For the second part, assume for a contradiction that there exist t ≥ 0 and x ∈ X 2,F \ {0} such that T t x 0.Then, T n x T n − t T t x 0 for every n ∈ N, n ≥ t and thus T n x n∈N ∈ F X .It follows that x ∈ X 1,F , which is not possible since x ∈ X 2,F \ {0}.Thus, T t | X 2,F is one-to-one, for all t ≥ 0. Proposition 4.2.If the pair E, F is admissible to the semigroup {T t } t≥0 , then for each f ∈ E X , there exists unique Proof.Let f ∈ E X and x ∈ X from Definition 3.11.For y x − P 1 x P 2 x we have that y ∈ X 2,F and p f n; y p f n; x − T n P 1 x.Since p f •; x ∈ F X and T n P 1 x ∈ F X , it follows that p f •; y ∈ F X .
To prove the uniqueness of y, suppose that there exists z ∈ X 2,F with the property and therefore z y.
The unique vector y ∈ X 2,F will be denoted by x f .Proposition 4.3.If the pair E, F is admissible to the semigroup {T t } t≥0 , there exists K > 0 such that Proof.We define the operator It is obvious that U is a linear operator.Now, we will show that it is also closed.
where f ∈ E X , y ∈ X 2,F , and g ∈ F X .For each n ∈ N, we take x n x f n ∈ X 2,F and we have that lim n → ∞ u n m g k , for all m ∈ N. On the other hand, which implies lim n → ∞ u n m p f m; y , for all m ∈ N. It follows that p f •; y g ∈ F X and by Proposition 4.2 we have y x f .Therefore Uf y; g .Hence, U is a closed linear operator, and by the Closed-Graph Theorem it is also bounded which means that there exists K > 0 such that and the proof is complete.
A simple and useful evaluation that results from aProposition 4.3 and Proposition 2.8 is given in the following remark.
With this intermediate result we are able to prove that the admissibility of E, F to an exponentially bounded semigroup {T t } t≥0 is a sufficient condition for a no past exponential dichotomy of {T t } t≥0 .The restriction over such a pair E, F is that E and F are not simultaneously the bounds of the chain of sequence Schäffer spaces in the sense of Proposition 2.8 .Theorem 4.5.Let E and F be sequence Schäffer spaces such that 1 / E or ∞ 0 / ⊂F and let {T t } t≥0 be an exponentially bounded semigroup.If E, F is admissible to {T t } t≥0 , then it has a no past exponential dichotomy and X 1,F X 1 .

Proof (Part I). The exponential decay of {T
Let x ∈ X 1,F and consider the sequence for all n ∈ N.For t ≥ 0, taking n t we have

4.10
Since the constant C 1 : Me ω K β E 0 /β F 0 does not depend on x, we can write down For n, m ∈ N and x ∈ X 1,F , we evaluate 12 which implies T n x n j 0 δ j ≤ C 1 p f •; 0 and therefore

4.13
By Proposition 4.3 we get that If ∞ 0 / ⊂F, then β F is not bounded and therefore there exists n 0 ∈ N * such that η : If ∞ 0 ⊂ F, then E / 1 .From Proposition 2.8, it follows that there exists h ∈ E \ 1 .Consider which is nondecreasing and lim n → ∞ γ n ∞.For n ∈ N and x ∈ X 1,F , the sequence has finite support, and thus g ∈ E X .Observing that g m ≤ C 1 x |h m | for every m ∈ N, we are led to the evaluation g E X ≤ L h E x .Also, 4.17 and thus we have p g m; 0 ≤ γ n T m x , for all m ∈ N. Since x ∈ X 1,F , T m x m∈N ∈ F X , we have that p g •; 0 ∈ F X .Taking m n in p g m; 0 ≤ K/β F 0 g E X , we can deduce that It follows that there exists n 0 ∈ N * such that η : In both cases, we obtained the existence of some n 0 ∈ N * and some constant η ∈ 0, 1 such that T n 0 x ≤ η x for all x ∈ X 1,F .Therefore, the semigroup {T t |X 1,F } t≥0 is exponentially stable see Remark 3.7 .Subsequently, there exist N 1 , ν 1 > 0 such that
Let n ∈ N, n 0 ∈ N * and x ∈ X 2,F \ {0}, and consider the sequence We have that f ∈ E X with f E X β E 0 .On the one hand, for all m ∈ N where y −x/ T n n 0 x ∈ X 2,F , while on the other hand

4.23
Taking now m n 0 we can write down

4.24
For t ≥ 0, we put Also, from 4.23 , taking C : 1 K β E 0 /β F 0 we have that Since n, n 0 , and x are randomly taken see the beginning of Part II , we have that

and therefore T n x / T n n
or equivalently,

4.29
Using the fact that p f •; y F X ≤ K f E X , we deduce that If ∞ 0 / ⊂F, then β F is not bounded and therefore there exists n 0 ∈ N * such that η : If ∞ 0 ⊂ F, then E / 1 and therefore there exists h ∈ E \ 1 .Consider γ as in 4.15 , and for n, n 0 ∈ N and x ∈ X 2,F \ {0} we define for all m ∈ N where z : γ n 0 −x/ T n n 0 x ∈ X 2,F , while on the other hand

4.33
From 4.32 and 4.33 , it follows that p g •; z has finite support, and thus it belongs to F X .Therefore, p g n; z ≤ K/β F 0 g E X and using that g E X ≤ C h E we obtain that Then, there exist n 0 ∈ N * and η > 1 such that

4.35
In both cases, we obtained the existence of some n 0 ∈ N * and some constant η > 1 satisfying the condition ii from Lemma 3.9, while the first condition is assured by 4.25 .Subsequently, there exist N 2 , ν 2 > 0 such that T t x ≥ N 2 e ν 2 t x , ∀t ≥ 0, x ∈ X 2,F .

4.36
If we take ν : min{ν 1 , ν 2 } > 0, all the conditions guaranteeing the existence of a no past exponential dichotomy are met.
Proof (Part III).We prove that X 1,F X 1 , no matter how we choose the sequence Schäffer space F. If x ∈ X 1,F , then T t x ≤ N 1 e −ν 1 t x , for all t ≥ 0, which implies x ∈ X 1 .
Conversely, let x ∈ X 1 , u ∈ X 1,F , and v ∈ X 2,F such that x u v.For every n ∈ N, we have that

4.37
If we suppose that v / 0, then ∞ contradicting the fact that x ∈ X 1 .It follows immediately that v 0 and x u ∈ X 1,F .
Remark 4.6.As we pointed out in the introduction, there is an extensive literature on the connection between admissibility and hyperbolicity or equivalently, exponential dichotomy .Latest there is known the equivalence between the admissibility of the pair p , q 1 ≤ p ≤ q ≤ ∞ and p, q / 1, ∞ and the hyperbolicity of a C 0 -semigroup {T t } t≥0 , when we assume a priori that the kernel of the splitting projection is T t -invariant and T t | Ker P are invertible.
For details we refer the reader to 21 .We try to extend this line of results in two directions.First, we do not assume a priori that T t | Ker P is invertible we do not even assume that Ker P is T t -invariant and still we succeed to prove that the admissibility of any pair of sequence Schäffer spaces implies the existence of a no past exponential dichotomy.Secondly, it is worth to note that the class of sequence Schäffer spaces is extremely reachable see, e.g., Examples 2.3 and 2.4 and this fact allows the reader to choose the "test sequences" in various ways and in the same time it does not force the "output" or "nice discretetime mild solutions" i.e., the solution of the inhomogeneous difference equation problem to stay in q X , as before.Moreover, this approach can provide interesting input spaces i.e., the spaces consisting in "test sequences" which are different from the classical p spaces we refer the reader to Example 2.6 .Also, it is worth to note that if there exists a pair of vector-valued sequence Schäffer spaces E, F , which is admissible to {T t } t≥0 , and with the property that 1 / E or ∞ 0 / ⊂F, then the subspace X 1,F which induces the no past exponential dichotomy is actually the regular stable subspace X 1 .If we would impose in addition that the complement of X 1 denoted by X 2 is also T t -invariant, then the aforementioned admissibility condition would imply that {T t } t≥0 extends automatically to a C 0 -group on X 2 , and thus we would get hyperbolicity for {T t } t≥0 see Theorem 4.12 below .Therefore, we can conclude that "admissibility" converts to "no past exponential dichotomy""admissibility" and "T t -invariance of X 2 " converts to "hyperbolicity".
The next example shows that the condition "E / 1 or ∞ 0 / ⊂F" in the statement of Theorem 4.5 is essential.
Example 4.7.Let X R and consider the semigroup {T t } t≥0 by T t I R for all t ≥ 0. If f ∈ 1 , then there exists x − ∞ k 0 f k ∈ R the series being absolutely convergent such that p f •; x ∈ ∞ 0 X .Therefore, the pair 1 , ∞ 0 is admissible to {T t } t≥0 , but one can easily check that {T t } t≥0 does not posses a no past exponential dichotomy.Theorem 4.12.Let E and F be two sequence Schäffer spaces such that 1 / E or ∞ 0 / ⊂F and let {T t } t≥0 be an exponentially bounded semigroup.If i there exists t 0 > 0 such that P 2 T t P 2 T t P 2 for all t ∈ 0, t 0 , ii E, F is admissible to T , then T t : X 2 → X 2 is invertible for each t ≥ 0. Therefore, {T t } t≥0 has an exponential dichotomy.
Proof.Let t > t 0 and consider n ∈ N and δ ∈ 0, t 0 such that t nt 0 δ.If x ∈ X 2 , we have that T t x T δ T t 0 n x ∈ X 2 .Therefore, the condition i assures that the operator T t : X 2 → X 2 is well defined with respect to the range for each t ≥ 0 and from Theorem 4.5 we have that {T t } t≥0 has a no past exponential dichotomy.Let t ≥ 0, n 0 t 1, y ∈ X 2 and consider the sequence which belongs to E X with f E X β E 0 T n 0 − t y .Then, according to Proposition 4.2, there exists a unique x ∈ X 2 such that p f •; x ∈ F X .Since for all n ∈ N, n ≥ n 0 , we have that T t x, y ∈ X 2 and p f n; x T n − t T t x − y ≥ N 2 e ν 2 n−t T t x − y , ∀n ≥ n 0 .

4.41
If we assume that T t x − y / 0, then lim n → ∞ p f n; x ∞ that contradicts the fact that p f •; x ∈ ∞ X .It follows that T t x y.We proved that T t | X 2 is onto, and since the one-to-one property was already proved in the Remark 4.1, we get the invertibility.
Since only the property of invertibility of operators on X 2 restricts the no past exponential dichotomy to be an exponential dichotomy, we completed the proof.
In what follows we try to answer concerns regarding the converse of what was obtained with Theorem 4.12.

4.45
For any n ∈ N, we have the following evaluation:

4.46
From the hypothesis and Remark 4.13 we have that f 4.47 and thus g : L k f exists as an element in F. From 4.46 , denoting C : N P 1 P 2 we have that ϕ n ≤ Cg n for all n ∈ N, and therefore, ϕ ∈ F X .Note that for k ≥ n, since T k x T k − n T n x, we have that If we keep the condition E ⊂ F and prove in some other setting that ϕ ∈ E, the argument presented previously still works.Such a new setting is given by LE ⊂ E the space E is invariant under the left shift , since the series defining the element g will be absolutely convergent in E.
In the following result we put all the pieces together to provide a necessary and sufficient condition for the exponential dichotomy of an exponentially bounded semigroup.Corollary 4.16.Let {T t } t≥0 be an exponentially bounded semigroup and let E and F be two sequence Schäffer spaces with the following properties: i there exists t 0 > 0 such that P 2 T t P 2 T t P 2 for all t ∈ 0, t 0 ; ii E ⊂ F; iii LE ⊂ E or LF ⊂ F; Then, {T t } t≥0 has an exponential dichotomy if and only if the pair E, F is admissible to {T t } t≥0 .
Proof.The necessity follows from Theorem 4.14 and Remark 4.15, while the sufficiency is proved by Theorems 4.5 and 4.12.
Example 4.17.Consider the semigroup acting on R, | • | defined by T t x e −t x.Clearly, {T t } t≥0 defines a C 0 -semigroup with T t x e −t |x| for all t ≥ 0 and x ∈ X.Therefore, {T t } t≥0 is exponentially stable thus it is exponentially dichotomic .
Consider the sequence f : N → R f n 1/ n 1 .We have that f ∈ 2 \ 1 , and for any x ∈ R,

4.50
We obtained that p f •; x / ∈ 1 for any x ∈ X.Thus, the pair 2 , 1 is not admissible to {T t } t≥0 .
This example shows that the condition E ⊂ F in Theorem 4.14 cannot be dropped.If the semigroup is exponentially stable, then X 2 {0}, and therefore, the invariance condition of either E or F under the left shift is not necessary.The sequence g : ∞ k 0 e −νk R k f exists in E and ϕ n ≤ g n for all n ∈ N. Therefore, ϕ ∈ E, and since ϕ p f •; 0 , E ⊂ F, we deduce that there exists x ∈ X such that p f •; 0 ∈ F X .Corollary 4.19.Let p, q ∈ 1, ∞ such that p ≤ q.If the semigroup {T t } t≥0 is exponentially dichotomic, then the pair p , q is admissible to {T t } t≥0 .
Proof.The spaces p 1 ≤ p ≤ ∞ are invariant under the left shift, and with p ≤ q we also have that p ⊂ q .Applying now Theorem 4.14 we obtain the previously statement.

T n − k f k e −n x n k 0 e
− n−k 1 k 1 .4.49 It follows that |p f n; x | ≥ 1/ n 1 − e −n |x|, for all n ∈ N.Then,

Corollary 4 . 18 .
Let E and F be two sequence Schäffer spaces such that E ⊂ F. If {T t } t≥0 is an exponentially stable semigroup, then the pair E, F is admissible to {T t } t≥0 .Proof.Exponential stability is an exponential dichotomy with P 1 I.With this remark, observe that the expression of ϕ in 4.44 reduces to

Corollary 4 . 20 .
Let Φ be a sequence Orlicz space.The semigroup {T t } t≥0 is exponentially dichotomic if and only if the pair Φ , Φ is admissible to {T t } t≥0 .Proof.It follows immediately from Theorem 4.14 and Corollary 4.11, since any Φ is invariant under the left shift.
and g ∈ E such that |f| ≤ |g|, then f ∈ E and f E ≤ g E .Remark 2.2.By s 1 and s 2 we have that any sequence with finite support is contained in any sequence Schäffer space, hence χ {0,1,...,n} ∈ E for any sequence Schäffer space E and n ∈ N. The third property is called the ideal property and will play a central role in our investigations.
Example 2.3.Common instances of sequence Schäffer spaces are the spaces of p-summable sequences, namely, for p ∈ 1, ∞ ,p 1,...,n} E , E n > 0, for all n ∈ N. Example 2.4.Other remarkable examples of sequence Schäffer spaces are the sequence Orlicz spaces.Let ϕ : R → 0, ∞ be a left continuous, nondecreasing function and not identically 0 or ∞ on 0, ∞ .The Young function attached to ϕ is defined by Φ t t 0 ϕ s ds, t ≥ 0. Consider Φ generator of the semigroup {T t } t≥0 .If {T t } t≥0 is a C 0 -semigroup, then A is closed and densely defined and {T t } t≥0 is exponentially bounded for details, see 19, Section 1.2 .It is clear that a classical solution for A; x with A being the infinitesimal generator of {T t } t≥0 exists only if x ∈ D A in which case T • x is the unique classical solution; otherwise it is said to be the mild solution of A; x .
It will be clear see Example 4.17 that for a semigroup {T t } t≥0 that has exponential dichotomy, not every pair E, F of sequence Schäffer spaces is admissible to {T t } t≥0 .Remark 4.13.If E and F are two sequence Schäffer spaces such that LE ⊂ F, then E ⊂ F and Lf F ≤ f F for all f ∈ E. Proof.Let f ∈ E.Then, Rf ∈ E and therefore f L Rf ∈ F and Lf ∈ F. Since, |RLf m | ≤ |f m | for each m ∈ N, we deduce that Let E and F be two sequence Schäffer spaces such that L E∪F ⊂ F. If the semigroup {T t } t≥0 has an exponential dichotomy, then the pair E, F is admissible to {T t } t≥0 .
series being absolutely convergent .It follows that there exists x ∈ X such that p f •; x ϕ ∈ F X .Remark 4.15.Let us examine more closely the condition L E ∪ F ⊂ F and the proof of the above theorem.The condition is in fact equivalent with LF ⊂ F and E ⊂ F see Remark 4.13 .