#### Abstract

We analyze the existence of (no past) exponential dichotomies
for a well-posed autonomous differential equation (that generates a *C*_{0}-semigroup ). 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 spaces, sequence Orlicz spaces, etc.) and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.

#### 1. 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 and its connection with the existence of bounded solutions of the equation , where the state space is a finite-dimensional Banach space and the operator-valued function is bounded and continuous in the strong operator topology. Relevant results concerning the extension of Perronโs problem in the more general framework of infinite-dimensional Banach spaces were obtained by Daleckij and Krein [2], Bellman [3], Massera and Schรคffer [4], and more recently by van Neerven [5], and van Minh et al. [6].

For the case of discrete-time systems, analogous results were firstly obtained by Li in [7]. In his paper, we remark the same central concern as in Perronโs work, but in other terms. In fact it was proposed that the inhomogeneous equation is responsible in some sense for the asymptotic behavior of solutions of the homogeneous equation. In this spirit, there were established connections between the condition that the inhomogeneous equation has some bounded solution for every bounded โsecond memberโ on the one hand and a certain form of conditional stability of the solutions of the homogeneous equation on the other. This idea was later extensively developed for discrete-time systems in the infinite-dimensional case by Coffman and Schรคffer [8] and Henry [9]. More recently, we have the papers of Ben-Artzi and Gohberg [10], Pinto [11], and LaSalle [12]. Applications of this โdiscrete-time theoryโ to stability theory of linear continuous-time systems in infinite-dimensional spaces have been presented by Przyluski and Rolewicz in [13].

The dichotomy of the autonomous equation , (assuming its well-posedness, i.e., generates a -semigroup ) consists in the existence of a bounded projection such that the solutions that start in decay to zero and the solutions that start in are unbounded. In the hypothesis that is -invariant and finite-dimensional, the existence of a dichotomy for implies that solutions starting in exist in backward time (or equivalently, extends to a -group on ). This observation is not necessarily true in the infinite-dimensional setting, but required in many researches (see, e.g., [6, 14, 15]). The novelty of our approach consists in the fact that we do not assume *a priori* that the operators are invertible (we do not even assume that is -invariant), and subsequently the unstable subspace is allowed to be infinite-dimensional. 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 previous condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical spaces, sequence Orlicz spaces, etc.), and since we use a discrete-time technique, we are not forced to require any continuity or measurability hypotheses on the trajectories of the one-parameter semigroup generated by the differential system. Also, it is worth to mention that from discrete-time conditions we get information about the continuous-time behavior of the solutions.

#### 2. Sequence Schรคffer Spaces

Let be the set of all nonnegative integers, , the set of all real numbers, and we will denote by the greatest integer less than or equal with . The linear space of all real-valued sequences is denoted by . Let be a real or complex Banach space and consider the linear space of all sequences . For an -valued sequence , we can associate the sequence defining for all . We also consider two linear operators defined by
known as the *right shift operator*, respectively, the *left shift operator*. A simple verification gives us and for , , for all . If , the characteristic function of will be denoted by and for the simplicity of notation put for each .

*Definition 2.1. *A Banach space is said to be a *sequence Schรคffer space* if and the following conditions hold:,if , then and ,if and such that , then and .

*Remark 2.2. *By and we have that any sequence with finite support is contained in any sequence Schรคffer space, hence for any sequence Schรคffer space and . The third property is called the *ideal property* and will play a central role in our investigations.

*Example 2.3. *Common instances of sequence Schรคffer spaces are the spaces of -summable sequences, namely, for ,
The subspace of , (often denoted by ) with the induced norm is another example of sequence Schรคffer space.

It is easy to check that (the space of all convergent sequences) is not a sequence Schรคffer space.

The spaces , , and occupy particularly important positions in the class of sequence Schรคffer spaces. For a sequence Schรคffer space, we will define the sequences by
which are both nondecreasing and , for all .

*Example 2.4. *Other remarkable examples of sequence Schรคffer spaces are the *sequence Orlicz spaces*. Let be a left continuous, nondecreasing function and not identically 0 or on . The Young function attached to is defined by . Consider
For the Banach space the conditions , , and are verified; hence is a sequence Schรคffer space.

For , taking we have that . Even is a sequence Orlicz space, obtained from for and for [16, Sectionโโ3].

*Remark 2.5. *Let . If , then .

Indeed, if , then , for all , which is equivalent with
Let and . Using the fact that is nondecreasing, we have that
which implies that
for all . Hence and

*Example 2.6. *Consider :
We claim that , no matter how we choose .

Indeed, and , for all , and using the aforementioned remark, our claim follows easilly. Also, since .

*Remark 2.7. *By simple computations we obtain that
for (with the convention ) and for the Orlicz sequence spaces,

For two Banach spaces and , we say that is continuously embedded in () if and there exists such that for all . For the following three propositions, proofs can be retrieved from [8, Sectionโโ3].

Proposition 2.8. *If is a sequence Schรคffer space, then with for all and for all .*

Proposition 2.9. *If and are sequence Schรคffer spaces, then if and only if .*

Proposition 2.10. *Let be a sequence Schรคffer space. The following characterizations hold:*(i)* is bounded if and only if and ; *(ii)* is bounded if and only if .*

For being a sequence Schรคffer space and being a Banach space, we consider and . To prove that is a Banach space see, for example, [17, Remarkโโ2.1] or [8, Lemmaโโ3.8]. The following properties of this space are simple verifications.

Proposition 2.11. *The space is a Banach space with the following properties:*(i)*if has finite support, then ;*(ii)*if , then and ;*(iii)*if and such that , then and .*

To prevent any further confusion, let us fix the notation for the class of bounded linear operators acting on .

#### 3. Semigroups of Bounded Linear Operators, Exponential Dichotomy, and Admissibility

Let be a Banach space, , a linear operator, and consider the (abstract) Cauchy problem : The function is said to be a classical solution for the previous Cauchy problem if it is continuously differentiable on and satisfies . A theorem of E. Hille states that if the resolvent set of is nonempty, then for every the Cauchy problem has a unique classical solution if and only if is the infinitesimal generator of a -semigroup. For the proof of this result we guide the reader to [18, Theoremโโ6.7, page 150] or [19, Theoremโโ1.3, page 102].

*Definition 3.1. *A family of bounded linear operators acting on is called a *semigroup* if (where is the identity operator on ), for all .If (ii) holds for any , then is called a *group*. If in addition of (i) and (ii), there exist such that for all ,then is said to be *exponentially bounded*.

If is a closed subspace of and is a (-)semigroup such that is -invariant for any , then the restrictions form a (-)semigroup called the subspace (-)semigroup on ([18, page 43]). A (-)semigroup can be extended to a (-)group if and only if there exists such that is invertible (see [18, page 80] or [19, Sectionโโ1.6]).

The linear operator defined by
is the *infinitesimal generator* of the semigroup . If is a -semigroup, then is closed and densely defined and is exponentially bounded (for details, see [19, Sectionโโ1.2]). It is clear that a classical solution for with being the infinitesimal generator of exists only if in which case is the unique classical solution; otherwise it is said to be the *mild solution* of .

A linear and bounded operator acting on a complex Banach space is said to be *hyperbolic* if (where denotes the unit circle in the complex plane and is the spectrum of ). The spectral Riesz projection for a hyperbolic operator is given by
The projection corresponds to the part of the spectrum of contained in the open unit disk . We note that the projection commutes with . Since
we obtain the spectral radius and also the operator is invertible with . Hence, if is hyperbolic, then there exist the constants such that, for all integers ,
(for the previous exposure we consulted [14, page 28]).

*Definition 3.2. *A semigroup is *hyperbolic* if there exists such that is an hyperbolic operator.

*Definition 3.3. *The semigroup has an *exponential dichotomy* (or that it is exponentially dichotomic) if there exist a projection (i.e., and ) and the constants such that (i), for all , (ii) is an isomorphism, for each , (iii), for all and , (iv), for all and .

The first condition in the previous definition expresses equivalently that and are both -invariant; the essence of (ii) is that can be extended to a group. The next result establishes the relation between hyperbolicity and exponential dichotomy for -semigroups on complex Banach spaces. For the proof we refer the reader to [14, Lemmaโโ2.15], [20, Theoremโโ1.1], or alternatively [15].

Proposition 3.4. *For a strongly continuous semigroup acting on a complex Banach space the following statements are equivalent. *(i)* is hyperbolic. *(ii)* has an exponential dichotomy. ** Moreover, if holds, then where , , and
**
is the spectral Riesz projection for that corresponds to . Also, if has an exponential dichotomy, then , for every .*

If generates an exponentially dichotomic -semigroup, then the differential equation has the property that the solutions starting from (resp., from ) decay exponentially for (resp., for ) 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 to exist for negative time, or in counterpart it forces the semigroup to be invertible on . 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 with an exponential blow-up in positive time. We will call the โexponential decay on and exponential blow-up on โ (both on positive time) behavior as no past exponential dichotomy.

*Definition 3.5. *The semigroup has a *no past exponential dichotomy* if there exist a projection and the constants such that (i), for all , (ii), for all , , (iii), for all , .

*Definition 3.6. *The semigroup has an *ordinary dichotomy* if there exist a projection and the constants such that (i), for all , (ii), for all , , (iii), for all , .

*Remark 3.7. *Note that if is one-to-one, then and thus the concept of (no past) exponential dichotomy overlaps the concept of *exponential stability*. Recall that is said to be exponentially stable if one of the following equivalent statements is true: (i)there exist such that for all ; (ii)there exist such that .

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 -semigroup *โhas a no past exponential dichotomy but is not exponentially dichotomic*. Let be a matrix with real entries whose spectrum is contained in the open left-half plane and consider the right shift semigroup on , given by
Then has a no past exponential dichotomy on with and . However, the restriction of on is not onto, and thus is not exponentially dichotomic.

Lemma 3.9. *Let be an exponentially bounded semigroup. If there exist , a vector subspace of , , , and such that *(i)*, for all and , *(ii)*, for all and , ** then there exist which satisfy , for all and .*

*Proof. *Let , and . Then, there exist and such that . Therefore, . We take to obtain
For , take to have . Put in order to write .

Lemma 3.10. *Let be an exponentially bounded semigroup. If there exist a vector subspace of , , , and such that *(i)*, for all and and *(ii)*, for all and , ** then there exist which satisfy , for all and .*

*Proof. *It is analogous with the proof of Lemma 3.9.

Consider the autonomous inhomogeneous abstract Cauchy problem :
If generates the -semigroup , and , then the function given by
is said to be the *mild solution* of the Cauchy problem . 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 ), we refer the reader to [19, Sectionโโ4.2].

The โtest functions methodโ (or โPerron's methodโ) was often used until now (see e.g., [1โ4]) to study properties of asymptotic behavior such as exponential dichotomy. According to Massera and Schรคffer [4] by โtest functions methodโ it is meant the relation between certain โtest functionsโ and โnice solutionsโ of the inhomogeneous equations : . The crudest expression of this method is the notion of *admissibility* of a pair of classes of functions both in (Massera and Schรคffer named these classes of functions as the class of โ*test functions*โ and the class of โ*nice solutions*โ and defined the pair to be admissible if for every โtest functionโ , the equation has a โnice solutionโ (see [4, Chapter 5, page 124]).

In this spirit, we set the expression of the mild solution of the equation in discrete-time to give the following definition of admissibility in terms of โ*test sequences*โ and โ*nice discrete-time mild solutions*โ. 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 : .

*Definition 3.11. *Let be sequence Schรคffer spaces. The pair is said to be *admissible* to if for each , there exists such that , where
for each .

#### 4. Main Results

In this section, for being a semigroup on the Banach space (with assuring the exponential boundedness if it is the case for to be exponentially bounded) and and being two sequence Schรคffer spaces, we denote which are obviously vector subspaces of .

*Hypothesis 1. *The vector subspace is closed and admits a closed complement; that is, there exists , a closed vector subspace, such that

We denote by the projection onto along and set (we will prove that in the case of a no past exponential dichotomy for , coincides always with and thus the -independent notation for projectors is consistent).

*Remark 4.1. *We have that and , for each and .

*Proof. *If and , then it is to see that for all , and since , it follows that .

For the second part, assume for a contradiction that there exist and such that . Then, for every , and thus . It follows that , which is not possible since . Thus, is one-to-one, for all .

Proposition 4.2. *If the pair is admissible to the semigroup , then for each , there exists unique such that .*

*Proof. *Let and from Definition 3.11. For we have that and . Since and , it follows that .

To prove the uniqueness of , suppose that there exists with the property . Since , we have that and therefore .

The unique vector will be denoted by .

Proposition 4.3. *If the pair is admissible to the semigroup , there exists such that
**
for all .*

*Proof. *We define the operator
It is obvious that is a linear operator. Now, we will show that it is also closed.

Let such that and
where , , and . For each , we take and . We have that and .

Since
we have that , for all . On the other hand,
which implies , for all . It follows that and by Proposition 4.2 we have . Therefore .

Hence, is a closed linear operator, and by the Closed-Graph Theorem it is also bounded which means that there exists 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.

*Remark 4.4. *If is admissible to the semigroup , then is admissible to and , for all and .

With this intermediate result we are able to prove that the admissibility of to an exponentially bounded semigroup is a sufficient condition for a no past exponential dichotomy of . The restriction over such a pair is that and are not simultaneously the bounds of the chain of sequence Schรคffer spaces (in the sense of Proposition 2.8).

Theorem 4.5. *Let and be sequence Schรคffer spaces such that or and let be an exponentially bounded semigroup. If is admissible to , then it has a no past exponential dichotomy and .*

*Proof (Part I). *The exponential decay of on .

Let and consider the sequence
which is in with . Observe that , for all . Then, and from we obtain , for all . For , taking we have
Since the constant does not depend on , we can write down
For and , we evaluate
which implies and therefore
By Proposition 4.3 we get that

If , then is not bounded and therefore there exists such that and , for all .

If , then . From Proposition 2.8, it follows that there exists . Consider
which is nondecreasing and . For and , the sequence
has finite support, and thus . Observing that for every , we are led to the evaluation . Also,
and thus we have , for all . Since , , we have that . Taking in , we can deduce that
It follows that there exists such that and , for all .

In both cases, we obtained the existence of some and some constant such that for all . Therefore, the semigroup is exponentially stable (see Remark 3.7). Subsequently, there exist such that

*Proof (Part II). *The exponential blow-up of on .

Let , and , and consider the sequence
We have that with . On the one hand,
for all (where ), while on the other hand
which implies that . Then, and
Taking now we can write down
For , we put to evaluate . Denoting we have that
Also, from (4.23), taking we have that
Since , and are randomly taken (see the beginning of *Part II*), we have that
and therefore . It follows that
or equivalently,
Using the fact that (4.29), we deduce that

If , then is not bounded and therefore there exists such that and for all and .

If , then and therefore there exists . Consider as in (4.15), and for and we define
Since for each , we have that with . Note that , for all . On the one hand,
for all (where ), while on the other hand
From (4.32) and (4.33), it follows that has finite support, and thus it belongs to . Therefore, and using that we obtain that
Thus, , for all and . Then, there exist and such that

In both cases, we obtained the existence of some and some constant satisfying the condition from Lemma 3.9, while the first condition is assured by (4.25). Subsequently, there exist such that

If we take , all the conditions guaranteeing the existence of a no past exponential dichotomy are met.

*Proof (Part III). *We prove that , no matter how we choose the sequence Schรคffer space . If , then , for all , which implies .

Conversely, let , , and such that . For every , we have that
If we suppose that , then contradicting the fact that . It follows immediately that and .

*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 and and the hyperbolicity of a -semigroup , when we assume *a priori* that the kernel of the splitting projection is -invariant and 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 is invertible (we do not even assume that is -invariant) and still we succeed to prove that the admissibility of any pair of sequence Schรคffer spaces implies the existence of *a no past exponential dichotomy.* Secondly, it is worth to note that the class of sequence Schรคffer spaces is extremely reachable (see e.g., Examples 2.3 and 2.4) and this fact allows the reader to choose the test sequencesโ in various ways and in the same time it does not force the โoutputโ or โnice discrete-time mild solutionsโ (i.e., the solution of the inhomogeneous difference equation problem) to stay in , as before. Moreover, this approach can provide interesting *input spaces* (i.e., the spaces consisting in "test sequences") which are different from the classical 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 Schรคffer spaces , which is admissible to , and with the property that or , then the subspace (which induces the no past exponential dichotomy) is actually the regular stable subspace . If we would impose in addition that the complement of (denoted by ) is also -invariant, then the aforementioned admissibility condition would imply that extends automatically to a -group on , and thus we would get hyperbolicity for (see Theorem 4.12 below). Therefore, we can conclude that โadmissibilityโ converts to โno past exponential dichotomyโโadmissibilityโ and โ-invariance of โ converts to โhyperbolicityโ.

The next example shows that the condition โ or โ in the statement of Theorem 4.5 is essential.

*Example 4.7. *Let and consider the semigroup by for all . If , then there exists (the series being absolutely convergent) such that . Therefore, the pair is admissible to , but one can easily check that does not posses a no past exponential dichotomy.

The above theorem shows that the space induces a no past exponential dichotomy for โan exponentially bounded semigroupโand that is actually . Concerning the hypothesis over and from the Theorem 4.5 we make one last remark.

*Remark 4.8. *In [8] there was introduced the order relation between pairs of Banach spaces: is said to be *stronger than * (or the later pair is *weaker than* the former) if and . In the case of sequence Schรคffer spaces the two inclusions with continuous injection are equivalent with the corresponding algebraic inclusion (Proposition 2.9); so the pair is stronger than if and , or equivalently (considering Proposition 2.8), and . Therefore, the hypothesis โ or โ can be replaced by, โthe pair is not stronger than the pair โ.

Dropping off the restriction โ or โ, the proof of Theorem 4.5 still provides useful information. More accurately, the admissibility of any pair of sequence Schรคffer spaces implies the existence of an ordinary dichotomy for the semigroup (in the sense of Definition 3.6).

Corollary 4.9. *Let be a pair sequence Schรคffer space and, be an exponentially bounded semigroup. If the pair is admissible to , then has an ordinary dichotomy; that is, there exist two constants such that
*

*Proof. *Note that in the proof of Theorem 4.5, we do not use the hypothesis โ or โ to prove the relations (4.11) and (4.25).

Corollary 4.10. *Let such that and let be an exponentially bounded semigroup. If the pair is admissible to , then has a no past exponential dichotomy.*

*Proof. *We have the equivalence (due to Proposition 2.10 and the fact that for each ): if and only if . Thus, given the hypothesis of Theorem 4.5 is satisfied and the conclusion is immediate.

Corollary 4.11. *Let be a sequence Orlicz space and let be an exponentially bounded semigroup. If the pair is admissible to , then has a no past exponential dichotomy.*

*Proof. *It follows from Theorem 4.5 and Example 2.4.

With Theorem 4.12 we prove that if we impose the -invariance of , we can deduce the invertibility of , thus obtaining the exponential dichotomy (or equivalently, hyperbolicity) for . Note that the first condition of the following theorem does not require that is invariant to all operators . However, we can prove that is a semigroup (exponentially bounded if is exponentially bounded); hence the invertibility of all its operators follows from the invertibility of just one of them.

Theorem 4.12. *Let and be two sequence Schรคffer spaces such that or and let be an exponentially bounded semigroup. If*(i)*there exists such that for all , *(ii)* is admissible to ,** then is invertible for each . Therefore, has an exponential dichotomy.*

*Proof. *Let and consider and such that . If , we have that . Therefore, the condition assures that the operator is well defined (with respect to the range) for each and from Theorem 4.5 we have that has a no past exponential dichotomy.

Let , , and consider the sequence
which belongs to with . Then, according to Proposition 4.2, there exists a unique such that . Since
for all , , we have that and
If we assume that , then that contradicts the fact that . It follows that . We proved that 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 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. It will be clear (see Example 4.17) that for a semigroup