We analyze the existence of (no past) exponential dichotomies for a well-posed autonomous differential equation (that generates a C0-semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0). 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 โ„“๐‘ 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 ๐‘ฅโ€ฒ=๐ด(๐‘ก)๐‘ฅ and its connection with the existence of bounded solutions of the equation ๐‘ฅโ€ฒ=๐ด(๐‘ก)๐‘ฅ+๐‘“(๐‘ก,๐‘ฅ), where the state space ๐‘‹ is a finite-dimensional Banach space and the operator-valued function ๐ด(โ‹…) is bounded and continuous in the strong operator topology. Relevant results concerning the extension of Perronโ€™s 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 Perronโ€™s 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 โ€œsecond 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 โ€œdiscrete-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 (๐ด), ๐‘ฅ๎…ž(๐‘ก)=๐ด๐‘ฅ(๐‘ก) (assuming its well-posedness, i.e., ๐ด generates a ๐ถ0-semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0) consists in the existence of a bounded projection ๐‘ƒ such that the solutions that start in ๐‘‹1=Im๐‘ƒ decay to zero and the solutions that start in ๐‘‹2=Im(๐ผโˆ’๐‘ƒ) are unbounded. In the hypothesis that ๐‘‹2 is ๐‘‡(๐‘ก)-invariant and finite-dimensional, the existence of a dichotomy for (๐ด) implies that solutions starting in ๐‘‹2 exist in backward time (or equivalently, {๐‘‡(๐‘ก)} extends to a ๐ถ0-group on ๐‘‹2). 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 ๐‘‡(๐‘ก)|๐‘‹2 are invertible (we do not even assume that ๐‘‹2 is ๐‘‡(๐‘ก)-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 โ„“๐‘ 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 โ„• be the set of all nonnegative integers, โ„•โˆ—=โ„•โงต{0}, โ„ the set of all real numbers, and we will denote by [๐‘ก] the greatest integer less than or equal with ๐‘ก. The linear space of all real-valued sequences ๐‘ โˆถโ„•โ†’โ„ is denoted by ๐’ฎ. Let (๐‘‹,โ€–โ‹…โ€–) be a real or complex Banach space and consider ๐’ฎ(๐‘‹) the linear space of all sequences ๐‘“โˆถโ„•โ†’๐‘‹. For an ๐‘‹-valued sequence ๐‘“โˆถโ„•โ†’๐‘‹, we can associate the sequence โ€–๐‘“โ€–โˆถโ„•โ†’โ„ defining โ€–๐‘“โ€–(๐‘›)=โ€–๐‘“(๐‘›)โ€– for all ๐‘›โˆˆโ„•. We also consider two linear operators ๐‘…,๐ฟโˆถ๐’ฎ(๐‘‹)โ†’๐’ฎ(๐‘‹) defined by ๎‚ป๐‘…๐‘“(๐‘›)=๐‘“(๐‘›โˆ’1),๐‘›โˆˆโ„•โˆ—,0,๐‘›=0,๐ฟ๐‘“(๐‘›)=๐‘“(๐‘›+1),(2.1) known as the right shift operator, respectively, the left shift operator. A simple verification gives us ๐ฟ๐‘…๐‘“=๐‘“ and ๐‘…๐ฟ๐‘“(๐‘›)=๐‘“(๐‘›) for ๐‘›โˆˆโ„•โˆ—, ๐‘…๐ฟ๐‘“(0)=0, for all ๐‘“โˆˆ๐’ฎ(๐‘‹). If ๐ดโŠ‚โ„•, the characteristic function of ๐ด will be denoted by ๐œ’๐ด and for the simplicity of notation put ๐›ฟ๐‘˜=๐œ’{๐‘˜} for each ๐‘˜โˆˆโ„•.

Definition 2.1. A Banach space (โ„ฐ,โ€–โ‹…โ€–โ„ฐ) is said to be a sequence Schรคffer space if โ„ฐโŠ‚๐’ฎ and the following conditions hold:(๐‘ 1)๐›ฟ0โˆˆโ„ฐ,(๐‘ 2)if ๐‘“โˆˆโ„ฐ, then ๐‘…๐‘“โˆˆโ„ฐ and โ€–๐‘…๐‘“โ€–โ„ฐ=โ€–๐‘“โ€–โ„ฐ,(๐‘ 3)if ๐‘“โˆˆ๐’ฎ and ๐‘”โˆˆโ„ฐ such that |๐‘“|โ‰ค|๐‘”|, then ๐‘“โˆˆโ„ฐ and โ€–๐‘“โ€–โ„ฐโ‰คโ€–๐‘”โ€–โ„ฐ.

Remark 2.2. By (๐‘ 1) and (๐‘ 2) we have that any sequence with finite support is contained in any sequence Schรคffer space, hence ๐œ’{0,1,โ€ฆ,๐‘›}โˆˆโ„ฐ for any sequence Schรคffer space โ„ฐ and ๐‘›โˆˆโ„•. 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 ๐‘-summable sequences, namely, for ๐‘โˆˆ[1,โˆž), โ„“๐‘=๎ƒฏ๐‘“โˆถโ„•โ†’โ„โˆถโˆž๎“๐‘˜=0||||๐‘“(๐‘˜)๐‘๎ƒฐ<โˆž,withthenormโ€–๐‘“โ€–๐‘=๎ƒฉโˆž๎“๐‘˜=0|๐‘“(๐‘˜)|๐‘๎ƒช1/๐‘,โ„“โˆž=๎‚ป๐‘“โˆถโ„•โ†’โ„โˆถsup๐‘›โˆˆโ„•||||๎‚ผ๐‘“(๐‘›)<โˆž,withthenormโ€–๐‘“โ€–โˆž=sup๐‘›โˆˆโ„•||||.๐‘“(๐‘›)(2.2) The subspace of โ„“โˆž, โ„“โˆž0={๐‘“โˆˆโ„“โˆžโˆถlim๐‘›โ†’โˆž๐‘“(๐‘›)=0} (often denoted by ๐‘0) with the induced norm is another example of sequence Schรคffer space.
It is easy to check that (๐‘,โ€–โ‹…โ€–โˆž) (the space of all convergent sequences) is not a sequence Schรคffer space.
The spaces โ„“1, โ„“โˆž, and โ„“โˆž0 occupy particularly important positions in the class of sequence Schรคffer spaces. For โ„ฐ a sequence Schรคffer space, we will define the sequences ๐›ผโ„ฐ,๐›ฝโ„ฐโˆˆ๐’ฎ by ๐›ผโ„ฐ๎ƒฏ(๐‘›)=inf๐ฟ>0โˆถ๐‘›๎“๐‘˜=0||||๐‘“(๐‘˜)โ‰ค๐ฟโ€–๐‘“โ€–โ„ฐ๎ƒฐ,๐›ฝ,โˆ€๐‘“โˆˆโ„ฐโ„ฐโ€–โ€–๐œ’(๐‘›)={0,1,โ€ฆ,๐‘›}โ€–โ€–โ„ฐ,(2.3) which are both nondecreasing and ๐›ฝโ„ฐ(๐‘›)>0, for all ๐‘›โˆˆโ„•.

Example 2.4. Other remarkable examples of sequence Schรคffer spaces are the sequence Orlicz spaces. Let ๐œ‘โˆถโ„โ†’[0,โˆž] be a left continuous, nondecreasing function and not identically 0 or โˆž on (0,โˆž). The Young function attached to ๐œ‘ is defined by โˆซฮฆ(๐‘ก)=๐‘ก0๐œ‘(๐‘ )๐‘‘๐‘ ,๐‘กโ‰ฅ0. Consider โ„“ฮฆ=๎ƒฏ๐‘“โˆˆ๐’ฎโˆถexists๐‘>0suchthat๐‘›๎“๐‘˜=0ฮฆ๎€ท๐‘||||๎€ธ๎ƒฐ๐‘“(๐‘˜)<โˆžwiththenormโ€–๐‘“โ€–ฮฆ๎ƒฏ=inf๐‘>0๐‘›๎“๐‘˜=0ฮฆ๎€ท๐‘โˆ’1||||๎€ธ๎ƒฐ๐‘“(๐‘˜)โ‰ค1(theLuxemburgnorm).(2.4) For the Banach space (โ„“ฮฆ,โ€–โ‹…โ€–ฮฆ) the conditions (๐‘ 1), (๐‘ 2), and (๐‘ 3) are verified; hence (โ„“ฮฆ,โ€–โ‹…โ€–ฮฆ) is a sequence Schรคffer space.
For 1โ‰ค๐‘<โˆž, taking ๐œ‘(๐‘ก)=๐‘๐‘ก๐‘โˆ’1 we have that (โ„“ฮฆ,โ€–โ‹…โ€–ฮฆ)โ‰ก(โ„“๐‘,โ€–โ‹…โ€–๐‘). Even โ„“โˆž is a sequence Orlicz space, obtained from ๐œ‘(๐‘ก)=0 for ๐‘กโˆˆ[0,1] and ๐œ‘(๐‘ก)=โˆž for ๐‘ก>1 [16, Sectionโ€‰โ€‰3].

Remark 2.5. Let ๐‘โˆˆ[1,โˆž). If (โ„“ฮฆ,โ€–โ‹…โ€–ฮฆ)=(โ„“๐‘,โ€–โ‹…โ€–๐‘), then lim๐‘กโ†’0(ฮฆ(๐‘ก)/๐‘ก๐‘)=1.
Indeed, if (โ„“ฮฆ,โ€–โ‹…โ€–ฮฆ)=(โ„“๐‘,โ€–โ‹…โ€–๐‘), then โ€–๐œ’{0,1,โ€ฆ,๐‘›}โ€–ฮฆ=โ€–๐œ’{0,1,โ€ฆ,๐‘›}โ€–๐‘, for all ๐‘›โˆˆโ„•, which is equivalent with ฮฆโˆ’1๎‚€1๎‚=๎‚€1๐‘›+1๎‚๐‘›+11/๐‘,โˆ€๐‘›โˆˆโ„•.(2.5) Let ๐‘ฅโˆˆ(0,1] and ๐‘š=[1/๐‘ฅ]โˆˆโ„•โˆ—. Using the fact that ฮฆโˆ’1 is nondecreasing, we have that ๎‚€1๎‚๐‘š+11/๐‘=ฮฆโˆ’1๎‚€1๎‚๐‘š+1โ‰คฮฆโˆ’1(๐‘ฅ)โ‰คฮฆโˆ’1๎‚€1๐‘š๎‚=๎‚€1๐‘š๎‚1/๐‘,(2.6) which implies that ๎‚ธ1([]๎‚น1/๐‘ฅ+1)๐‘ฅ1/๐‘โ‰คฮฆโˆ’1(๐‘ฅ)๐‘ฅ1/๐‘โ‰ค๎‚ธ1๐‘ฅ[]๎‚น1/๐‘ฅ1/๐‘,(2.7) for all ๐‘ฅโˆˆ(0,1]. Hence lim๐‘ฅโ†’0ฮฆโˆ’1(๐‘ฅ)๐‘ฅโˆ’(1/๐‘)=1 and lim๐‘ขโ†’0ฮฆ(๐‘ข)๐‘ข๐‘=lim๐‘ขโ†’01๎€บฮฆโˆ’1(ฮฆ(๐‘ข))/(ฮฆ(๐‘ข))1/๐‘๎€ป๐‘=1..(2.8)

Example 2.6. Consider ๐œ‘โˆถโ„+โ†’โ„+: ๐œ‘(๐‘ก)=โˆž๎“๐‘š=1๐‘šโˆš๐‘ก๐‘š2๎€œ,ฮฆ(๐‘ก)=๐‘ก0๐œ‘(๐‘ )๐‘‘๐‘ =โˆž๎“๐‘š=1๐‘ก1+1/๐‘š๐‘š(๐‘š+1).(2.9) We claim that โ„“ฮฆโ‰ โ„“๐‘, no matter how we choose ๐‘โˆˆ[1,โˆž).
Indeed, lim๐‘กโ†’0(ฮฆ(๐‘ก)/๐‘ก)=0 and lim๐‘กโ†’0(ฮฆ(๐‘ก)/๐‘ก๐‘)=โˆž, for all ๐‘โˆˆ(1,โˆž), and using the aforementioned remark, our claim follows easilly. Also, โ„“ฮฆโ‰ โ„“โˆž since ๐œ’โ„•โˆˆโ„“โˆžโงตโ„“ฮฆ.

Remark 2.7. By simple computations we obtain that ๐›ผโ„“๐‘(๐‘›)=(๐‘›+1)1โˆ’1/๐‘,๐›ฝโ„“๐‘(๐‘›)=(๐‘›+1)1/๐‘(2.10) for 1โ‰ค๐‘โ‰คโˆž (with the convention 1/โˆž=0) and for the Orlicz sequence spaces, ๐›ผโ„“ฮฆ(๐‘›)=(๐‘›+1)ฮฆโˆ’1๎‚€1๎‚๐‘›+1,๐›ฝโ„“ฮฆ1(๐‘›)=ฮฆโˆ’1(1/(๐‘›+1)).(2.11)
For two Banach spaces (๐ต1,โ€–โ‹…โ€–1) and (๐ต2,โ€–โ‹…โ€–2), we say that ๐ต1 is continuously embedded in ๐ต2 (๐ต1โ†ช๐ต2) if ๐ต1โŠ‚๐ต2 and there exists ๐‘>0 such that โ€–๐‘“โ€–2โ‰ค๐‘โ€–๐‘“โ€–1 for all ๐‘“โˆˆ๐ต1. For the following three propositions, proofs can be retrieved from [8, Sectionโ€‰โ€‰3].

Proposition 2.8. If (โ„ฐ,โ€–โ‹…โ€–โ„ฐ) is a sequence Schรคffer space, then โ„“1โ†ชโ„ฐโ†ชโ„“โˆž with ๐›ฝโ„ฐ(0)โ€–๐‘“โ€–โˆžโ‰คโ€–๐‘“โ€–โ„ฐ for all ๐‘“โˆˆโ„ฐ and โ€–๐‘“โ€–โ„ฐโ‰ค๐›ฝโ„ฐ(0)โ€–๐‘“โ€–1 for all ๐‘“โˆˆโ„“1.

Proposition 2.9. If (โ„ฐ,โ€–โ‹…โ€–โ„ฐ) and (โ„ฑ,โ€–โ‹…โ€–โ„ฑ) are sequence Schรคffer spaces, then โ„ฐโ†ชโ„ฑ if and only if โ„ฐโŠ‚โ„ฑ.

Proposition 2.10. Let (โ„ฐ,โ€–โ‹…โ€–โ„ฐ) be a sequence Schรคffer space. The following characterizations hold:(i)๐›ผโ„ฐ is bounded if and only if โ„ฐ=โ„“1 and โ€–โ‹…โ€–1โˆผโ€–โ‹…โ€–โ„ฐ; (ii)๐›ฝโ„ฐ is bounded if and only if โ„“โˆž0โŠ‚โ„ฐ.

For (โ„ฐ,โ€–โ‹…โ€–โ„ฐ) being a sequence Schรคffer space and ๐‘‹ being a Banach space, we consider โ„ฐ(๐‘‹)={๐‘“โˆˆ๐’ฎ(๐‘‹)โˆถโ€–๐‘“โ€–โˆˆโ„ฐ} and โ€–๐‘“โ€–โ„ฐ(๐‘‹)=โ€–โ€–๐‘“โ€–โ€–๐ธ. To prove that (โ„ฐ(๐‘‹),โ€–โ‹…โ€–โ„ฐ(๐‘‹)) is a Banach space see, for example, [17, Remarkโ€‰โ€‰2.1] or [8, Lemmaโ€‰โ€‰3.8]. The following properties of this space are simple verifications.

Proposition 2.11. The space (โ„ฐ(๐‘‹),โ€–โ‹…โ€–โ„ฐ(๐‘‹)) is a Banach space with the following properties:(i)if ๐‘“โˆˆ๐’ฎ(๐‘‹) has finite support, then ๐‘“โˆˆโ„ฐ(๐‘‹);(ii)if ๐‘“โˆˆโ„ฐ(๐‘‹), then ๐‘…๐‘“โˆˆโ„ฐ(๐‘‹) and โ€–๐‘…๐‘“โ€–๐ธ(๐‘‹)=โ€–๐‘“โ€–โ„ฐ(๐‘‹);(iii)if ๐‘“โˆˆ๐’ฎ(๐‘‹) and ๐‘”โˆˆโ„ฐ(๐‘‹) such that โ€–๐‘“โ€–โ‰คโ€–๐‘”โ€–, then ๐‘“โˆˆโ„ฐ(๐‘‹) and โ€–๐‘“โ€–โ„ฐ(๐‘‹)โ‰คโ€–๐‘”โ€–โ„ฐ(๐‘‹).

To prevent any further confusion, let us fix the notation โ„ฌ(๐‘‹) for the class of bounded linear operators acting on ๐‘‹.

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

Let ๐‘‹ be a Banach space, ๐‘ฅ0โˆˆ๐‘‹, ๐ดโˆถ๐ท(๐ด)โŠ‚๐‘‹โ†’๐‘‹ a linear operator, and consider the (abstract) Cauchy problem (๐ด;๐‘ฅ0): ๐‘ฅฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก),๐‘กโ‰ฅ0,(0)=๐‘ฅ0.(3.1) The function ๐‘ขโˆถ[0,โˆž)โ†’๐ท(๐ด) is said to be a classical solution for the previous Cauchy problem if it is continuously differentiable on [0,โˆž) and satisfies (๐ด;๐‘ฅ0). A theorem of E. Hille states that if the resolvent set of ๐ด is nonempty, then for every ๐‘ฅโˆˆ๐ท(๐ด) the Cauchy problem (๐ด;๐‘ฅ) has a unique classical solution if and only if ๐ด is the infinitesimal generator of a ๐ถ0-semigroup. For the proof of this result we guide the reader to [18, Theoremโ€‰โ€‰6.7, page 150] or [19, Theoremโ€‰โ€‰1.3, page 102].

Definition 3.1. A family of bounded linear operators {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 acting on ๐‘‹ is called a semigroup if (i)๐‘‡(0)=๐ผ (where ๐ผ is the identity operator on ๐‘‹), (ii)๐‘‡(๐‘ก+๐‘ )=๐‘‡(๐‘ก)๐‘‡(๐‘ ) for all ๐‘ก,๐‘ โ‰ฅ0.If (ii) holds for any ๐‘ก,๐‘ โˆˆโ„, then {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is called a group. If in addition of (i) and (ii), there exist ๐‘€,๐œ”>0 such that (iii)โ€–๐‘‡(t)โ€–โ‰ค๐‘€๐‘’๐œ”๐‘ก for all ๐‘กโ‰ฅ0,then {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is said to be exponentially bounded.

If ๐‘Œ is a closed subspace of ๐‘‹ and {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is a (๐ถ0-)semigroup such that ๐‘Œ is ๐‘‡(๐‘ก)-invariant for any ๐‘กโ‰ฅ0, then the restrictions ๐‘‡(๐‘ก)|๐‘Œ form a (๐ถ0-)semigroup called the subspace (๐ถ0-)semigroup on ๐‘Œ ([18, page 43]). A (๐ถ0-)semigroup can be extended to a (๐ถ0-)group if and only if there exists ๐‘ก0>0 such that ๐‘‡(๐‘ก0) is invertible (see [18, page 80] or [19, Sectionโ€‰โ€‰1.6]).

The linear operator defined by ๎‚ป๐ท(๐ด)=๐‘ฅโˆˆ๐‘‹โˆถlim๐‘กโ†“0๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฅ๐‘ก๎‚ผ,exists๐ด๐‘ฅ=lim๐‘กโ†“0๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฅ๐‘กfor๐‘ฅโˆˆ๐ท(๐ด)(3.2) is the infinitesimal generator of the semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0. If {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is a ๐ถ0-semigroup, then ๐ด is closed and densely defined and {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is exponentially bounded (for details, see [19, Sectionโ€‰โ€‰1.2]). It is clear that a classical solution for (๐ด;๐‘ฅ) with ๐ด being the infinitesimal generator of {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 exists only if ๐‘ฅโˆˆ๐ท(๐ด) in which case ๐‘‡(โ‹…)๐‘ฅ is the unique classical solution; otherwise it is said to be the mild solution of (๐ด;๐‘ฅ).

A linear and bounded operator ๐‘‡ acting on a complex Banach space is said to be hyperbolic if ๐œŽ(๐‘‡)โˆฉฮ“=โˆ… (where ฮ“={๐‘งโˆˆโ„‚โˆถ|๐‘ง|=1} denotes the unit circle in the complex plane and ๐œŽ(๐‘‡) is the spectrum of ๐‘‡). The spectral Riesz projection ๐‘ƒ for a hyperbolic operator ๐‘‡ is given by1๐‘ƒ=๎€œ2๐œ‹๐‘–ฮ“(๐œ†๐ผโˆ’๐‘‡)โˆ’1๐‘‘๐œ†.(3.3) The projection corresponds to the part of the spectrum of ๐‘‡ contained in the open unit disk ๐ท(0,1). We note that the projection ๐‘ƒ commutes with ๐‘‡. Since ๐œŽ๎€ท๐‘‡||Im๐‘ƒ๎€ธ๎‚€๐‘‡||=๐œŽ(๐‘‡)โˆฉ๐ท(0,1),๐œŽIm(๐ผโˆ’๐‘ƒ)๎‚=๐œŽ(๐‘‡)โˆฉ(โ„‚โงต๐ท(0,1)),(3.4) we obtain the spectral radius ๐‘Ÿ(๐‘‡|Im๐‘ƒ)<1 and also the operator ๐‘‡|Im(๐ผโˆ’๐‘ƒ) is invertible with ๐‘Ÿ(๐‘‡โˆ’1|Im(๐ผโˆ’๐‘ƒ))<1. Hence, if ๐‘‡ is hyperbolic, then there exist the constants ๐‘,๐œˆ>0 such that, for all integers ๐‘›โ‰ฅ0, โ€–๐‘‡๐‘›๐‘ฅโ€–โ‰ค๐‘๐‘’โˆ’๐œˆ๐‘›โ€–๐‘‡โ€–๐‘ฅโ€–,โˆ€๐‘ฅโˆˆIm๐‘ƒ,๐‘›๐‘ฅโ€–โ‰ฅ๐‘โˆ’1๐‘’๐œˆ๐‘›โ€–๐‘ฅโ€–,โˆ€๐‘ฅโˆˆIm(๐ผโˆ’๐‘ƒ)(3.5) (for the previous exposure we consulted [14, page 28]).

Definition 3.2. A semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is hyperbolic if there exists ๐‘ก0>0 such that ๐‘‡(๐‘ก0) is an hyperbolic operator.

Definition 3.3. The semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an exponential dichotomy (or that it is exponentially dichotomic) if there exist a projection ๐‘ƒ (i.e., ๐‘ƒโˆˆโ„ฌ(๐‘‹) and ๐‘ƒ2=๐‘ƒ) and the constants ๐‘,๐œˆ>0 such that (i)๐‘ƒ๐‘‡(๐‘ก)=๐‘‡(๐‘ก)๐‘ƒ, for all ๐‘กโ‰ฅ0, (ii)๐‘‡(๐‘ก)|Ker๐‘ƒโˆถKer๐‘ƒโ†’Ker๐‘ƒ is an isomorphism, for each ๐‘กโ‰ฅ0, (iii)โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘๐‘’โˆ’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0 and ๐‘ฅโˆˆIm๐‘ƒ, (iv)โ€–๐‘‡(๐‘ก)โˆ’1|Ker๐‘ƒ๐‘ฅโ€–โ‰ค๐‘๐‘’โˆ’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0 and ๐‘ฅโˆˆKer๐‘ƒ.

The first condition in the previous definition expresses equivalently that Im๐‘ƒ and Ker๐‘ƒ are both ๐‘‡(๐‘ก)-invariant; the essence of (ii) is that {๐‘‡(๐‘ก)|Ker๐‘ƒ}๐‘กโ‰ฅ0 can be extended to a group. The next result establishes the relation between hyperbolicity and exponential dichotomy for ๐ถ0-semigroups on complex Banach spaces. For the proof we refer the reader to [14, Lemmaโ€‰โ€‰2.15], [20, Theoremโ€‰โ€‰1.1], or alternatively [15].

Proposition 3.4. For a strongly continuous semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 acting on a complex Banach space the following statements are equivalent. (i){๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is hyperbolic. (ii){๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an exponential dichotomy. Moreover, if (i) holds, then ๐‘‹=๐‘‹1โŠ•๐‘‹2 where ๐‘‹1=Im๐‘ƒ, ๐‘‹2=Im(๐ผโˆ’๐‘ƒ), and 1๐‘ƒ=๎€œ2๐œ‹๐‘–ฮ“๎€ท๎€ท๐‘ก๐œ†๐ผโˆ’๐‘‡0๎€ธ๎€ธโˆ’1๐‘‘๐œ†(3.6) is the spectral Riesz projection for ๐‘‡(๐‘ก0) that corresponds to ๐œŽ(๐‘‡(๐‘ก0))โˆฉฮ“=โˆ…. Also, if {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an exponential dichotomy, then ๐œŽ(๐‘‡(๐‘ก))โˆฉฮ“=โˆ…, for every ๐‘กโ‰ฅ0.

If ๐ด generates an exponentially dichotomic ๐ถ0-semigroup, then the differential equation ฬ‡๐‘ฅ=๐ด๐‘ฅ has the property that the solutions ๐‘ฅ(โ‹…) starting from ๐‘‹1 (resp., from ๐‘‹2) decay exponentially for ๐‘ก>0 (resp., for ๐‘ก<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 ๐‘‹2 to exist for negative time, or in counterpart it forces the semigroup to be invertible on ๐‘‹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 ๐‘‹2 with an exponential blow-up in positive time. We will call the โ€œexponential decay on ๐‘‹1 and exponential blow-up on ๐‘‹2โ€ (both on positive time) behavior as no past exponential dichotomy.

Definition 3.5. The semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has a no past exponential dichotomy if there exist a projection ๐‘ƒ and the constants ๐‘1,๐‘2,๐œˆ>0 such that (i)๐‘ƒ๐‘‡(๐‘ก)๐‘ƒ=๐‘‡(๐‘ก)๐‘ƒ, for all ๐‘กโ‰ฅ0, (ii)โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘1๐‘’โˆ’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0, ๐‘ฅโˆˆIm๐‘ƒ, (iii)โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐‘2๐‘’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0, ๐‘ฅโˆˆKer๐‘ƒ.

Definition 3.6. The semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an ordinary dichotomy if there exist a projection ๐‘ƒ and the constants ๐‘1,๐‘2>0 such that (i)๐‘ƒ๐‘‡(๐‘ก)๐‘ƒ=๐‘‡(๐‘ก)๐‘ƒ, for all ๐‘กโ‰ฅ0, (ii)โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘1โ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0, ๐‘ฅโˆˆIm๐‘ƒ, (iii)โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐‘2โ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0, ๐‘ฅโˆˆKer๐‘ƒ.

Remark 3.7. Note that if ๐‘ƒ is one-to-one, then Im๐‘ƒ=๐‘‹ and thus the concept of (no past) exponential dichotomy overlaps the concept of exponential stability. Recall that {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is said to be exponentially stable if one of the following equivalent statements is true: (i)there exist ๐‘,๐œˆ>0 such that โ€–๐‘‡(๐‘ก)โ€–โ‰ค๐‘๐‘’โˆ’๐œˆ๐‘ก for all ๐‘กโ‰ฅ0; (ii)there exist ๐‘ก0>0 such that โ€–๐‘‡(๐‘ก0)โ€–<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 ๐ถ0-semigroup {๐‘†(๐‘ก)}๐‘กโ‰ฅ0โ€‰has a no past exponential dichotomy but is not exponentially dichotomic. Let ๐ด be a ๐‘ร—๐‘ matrix with real entries whose spectrum is contained in the open left-half plane and consider the right shift semigroup on ๐ฟ1(โ„+,โ„), given by ๎‚ป๐‘‡(๐‘ก)๐‘“(๐‘ )=0,0โ‰ค๐‘ <๐‘ก,๐‘“(๐‘ โˆ’๐‘ก),๐‘ โ‰ฅ๐‘ก.(3.7) Then ๐‘†(๐‘ก)=๐‘’๐‘ก๐ดโŠ•๐‘’๐‘ก๐‘‡(๐‘ก),๐‘†(๐‘ก)(๐‘ฅ,๐‘“)=(๐‘’๐‘ก๐ด๐‘ฅ,๐‘’๐‘ก๐‘‡(๐‘ก)๐‘“) has a no past exponential dichotomy on ๐‘‹=โ„๐‘โŠ•๐ฟ1(โ„+,โ„) with ๐‘‹1={(๐‘ฅ,0)โˆถ๐‘ฅโˆˆโ„๐‘} and ๐‘‹2={(0,๐‘“)โˆถ๐‘“โˆˆ๐ฟ1(โ„+,โ„)}. However, the restriction of ๐‘†(๐‘ก) on ๐‘‹2 is not onto, and thus {๐‘†(๐‘ก)}๐‘กโ‰ฅ0 is not exponentially dichotomic.

Lemma 3.9. Let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If there exist ๐‘Œ, a vector subspace of ๐‘‹, ๐‘›0โˆˆโ„•โˆ—, ๐ป>0, and ๐œ‚>1 such that (i)โ€–T(๐‘›)๐‘ฅโ€–โ‰ฅ๐ปโ€–๐‘ฅโ€–, for all ๐‘›โˆˆ{0,1,โ€ฆ,๐‘›0} and ๐‘ฅโˆˆ๐‘Œ, (ii)โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–โ‰ฅ๐œ‚โ€–๐‘‡(๐‘›)๐‘ฅโ€–, for all ๐‘›โˆˆโ„• and ๐‘ฅโˆˆ๐‘Œ, then there exist ๐‘,๐œˆ>0 which satisfy โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐‘๐‘’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘ฅโˆˆ๐‘Œ and ๐‘กโ‰ฅ0.

Proof. Let ๐‘ฅโˆˆ๐‘Œ, and ๐‘›โˆˆโ„•โˆ—. Then, there exist ๐‘˜โˆˆโ„• and ๐‘Ÿโˆˆ{0,1,โ€ฆ,๐‘›0โˆ’1} such that ๐‘›=๐‘˜๐‘›0+๐‘Ÿ. Therefore, โ€–๐‘‡(๐‘›)๐‘ฅโ€–=โ€–๐‘‡(๐‘˜๐‘›0+๐‘Ÿ)๐‘ฅโ€–โ‰ฅ๐œ‚๐‘˜โ€–๐‘‡(๐‘Ÿ)๐‘ฅโ€–โ‰ฅ๐ป๐œ‚๐‘˜โ€–๐‘ฅโ€–. We take ๐œˆโˆถ=(1/๐‘›0)ln๐œ‚>0 to obtain โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ฅ๐ป๐‘’๐œˆ๐‘˜๐‘›0โ€–๐‘ฅโ€–=๐ป๐‘’โˆ’๐œˆ๐‘Ÿ๐‘’๐œˆ(๐‘˜๐‘›0+๐‘Ÿ)โ€–๐‘ฅโ€–>๐ป๐‘’โˆ’๐œˆ๐‘›0e๐œˆ๐‘›โ€–๐‘ฅโ€–,โˆ€๐‘›โˆˆโ„•.(3.8) For ๐‘กโ‰ฅ0, take ๐‘›=[๐‘ก]+1 to have ๐‘€๐‘’๐œ”โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅโ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ฅ๐ป๐‘’โˆ’๐œˆ๐‘›0๐‘’๐œˆ๐‘›โ€–๐‘ฅโ€–. Put ๐‘โˆถ=๐ป๐‘€โˆ’1๐‘’โˆ’(๐œˆ๐‘›0+๐œ”)>0 in order to write โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐‘๐‘’๐œˆ๐‘กโ€–๐‘ฅโ€–.

Lemma 3.10. Let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If there exist ๐‘Œ a vector subspace of ๐‘‹, ๐‘›0โˆˆโ„•โˆ—, ๐ป>0, and ๐œ‚โˆˆ(0,1) such that (i)โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ค๐ปโ€–๐‘ฅโ€–, for all ๐‘›โˆˆ{0,1,โ€ฆ,๐‘›0} and ๐‘ฅโˆˆ๐‘Œ and (ii)โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–โ‰ค๐œ‚โ€–๐‘‡(๐‘›)๐‘ฅโ€–, for all ๐‘›โˆˆโ„• and ๐‘ฅโˆˆ๐‘Œ, then there exist ๐‘,๐œˆ>0 which satisfy โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘๐‘’โˆ’๐œˆ๐‘กโ€–๐‘ฅโ€–, for all ๐‘ฅโˆˆ๐‘Œ and ๐‘กโ‰ฅ0.

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

Consider the autonomous inhomogeneous abstract Cauchy problem (๐ด,๐‘“;x0): ๐‘ฅฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๐‘“(๐‘ก),๐‘กโ‰ฅ0,(0)=๐‘ฅ0.(3.9) If ๐ด generates the ๐ถ0-semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, ๐‘ฅโˆˆ๐‘‹ and ๐‘“โˆˆ๐ฟ1loc(โ„+,๐‘‹), then the function ๐‘ขโˆถโ„+โ†’๐‘‹ given by ๎€œ๐‘ข(๐‘ก)=๐‘‡(๐‘ก)๐‘ฅ+๐‘ก0๐‘‡(๐‘กโˆ’๐‘ )๐‘“(๐‘ )๐‘‘๐‘ (3.10) is said to be the mild solution of the Cauchy problem (๐ด,๐‘“;๐‘ฅ). 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 (๐ด,๐‘“;๐‘ฅ)), we refer the reader to [19, Sectionโ€‰โ€‰4.2].

The โ€œtest functions methodโ€ (or โ€œPerron's methodโ€) was often used until now (see e.g., [1โ€“4]) to study properties of asymptotic behavior such as exponential dichotomy. According to Massera and Schรคffer [4] by โ€œtest functions methodโ€ it is meant the relation between certain โ€œtest functionsโ€ ๐‘“ and โ€œnice solutionsโ€ of the inhomogeneous equations (๐ด,๐‘“): ๐‘ฅ๎…ž=๐ด๐‘ฅ+๐‘“. The crudest expression of this method is the notion of admissibility of a pair of classes of functions both in ๐ฟ1loc(โ„+,๐‘‹) (Massera and Schรคffer named these classes of functions as the class of โ€œtest functionsโ€ and the class of โ€œnice solutionsโ€ and defined the pair to be admissible if for every โ€œtest functionโ€ ๐‘“, the equation (๐ด,๐‘“) has a โ€œnice solutionโ€ (see [4, Chapter 5, page 124]).

In this spirit, we set the expression of the mild solution of the equation (๐ด,๐‘“) in discrete-time to give the following definition of admissibility in terms of โ€œtest sequencesโ€ and โ€œnice discrete-time mild solutionsโ€. 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 (๐ด): ๐‘ฅโ€ฒ=๐ด๐‘ฅ.

Definition 3.11. Let โ„ฐ,โ„ฑ be sequence Schรคffer spaces. The pair (โ„ฐ,โ„ฑ) is said to be admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 if for each ๐‘“โˆˆโ„ฐ(๐‘‹), there exists ๐‘ฅโˆˆ๐‘‹ such that ๐‘๐‘“(โ‹…;๐‘ฅ)โˆˆโ„ฑ(๐‘‹), where ๐‘๐‘“(๐‘›;๐‘ฅ)=๐‘‡(๐‘›)๐‘ฅ+๐‘›๎“๐‘˜=0๐‘‡(๐‘›โˆ’๐‘˜)๐‘“(๐‘˜),(3.11) for each ๐‘›โˆˆโ„•.

4. Main Results

In this section, for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 being a semigroup on the Banach space ๐‘‹ (with ๐‘€,๐œ” assuring the exponential boundedness if it is the case for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 to be exponentially bounded) and โ„ฐ and โ„ฑ being two sequence Schรคffer spaces, we denote ๐‘‹1,โ„ฑ=๎€ฝ๐‘ฅโˆˆ๐‘‹โˆถ(๐‘‡(๐‘›)๐‘ฅ)๐‘›โˆˆโ„•๎€พ,๐‘‹โˆˆโ„ฑ(๐‘‹)1=๎‚ป๐‘ฅโˆˆ๐‘‹โˆถlim๐‘กโ†’โˆž๎‚ผ,๐‘‡(๐‘ก)๐‘ฅ=0(4.1) which are obviously vector subspaces of ๐‘‹.

Hypothesis 1. The vector subspace ๐‘‹1,โ„ฑ is closed and admits a closed complement; that is, there exists ๐‘‹2,โ„ฑ, a closed vector subspace, such that ๐‘‹=๐‘‹1,โ„ฑโŠ•๐‘‹2,โ„ฑ.(4.2)
We denote by ๐‘ƒ1 the projection onto ๐‘‹1,โ„ฑ along ๐‘‹2,โ„ฑ and set ๐‘ƒ2=๐ผโˆ’๐‘ƒ1 (we will prove that in the case of a no past exponential dichotomy for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, ๐‘‹1,โ„ฑ coincides always with ๐‘‹1 and thus the โ„ฑ-independent notation for projectors is consistent).

Remark 4.1. We have that ๐‘‡(๐‘ก)๐‘‹1,โ„ฑโŠ‚๐‘‹1,โ„ฑ and ๐‘‡(๐‘ก)๐‘ฅโ‰ 0, for each ๐‘ฅโˆˆ๐‘‹2,โ„ฑโงต{0} and ๐‘กโ‰ฅ0.

Proof. If ๐‘ฅโˆˆ๐‘‹1,โ„ฑ and ๐‘กโ‰ฅ0, then it is to see that โ€–๐‘‡(๐‘›)๐‘‡(๐‘ก)๐‘ฅโ€–โ‰คโ€–๐‘‡(๐‘ก)โ€–โ€–๐‘‡(๐‘›)๐‘ฅโ€– for all ๐‘›โˆˆโ„•, and since (โ€–๐‘‡(๐‘›)๐‘ฅโ€–)๐‘›โˆˆโ„•โˆˆโ„ฑ, it follows that ๐‘‡(๐‘ก)๐‘ฅโˆˆ๐‘‹1,โ„ฑ.
For the second part, assume for a contradiction that there exist ๐‘กโ‰ฅ0 and ๐‘ฅโˆˆ๐‘‹2,โ„ฑโงต{0} such that ๐‘‡(๐‘ก)๐‘ฅ=0. Then, ๐‘‡(๐‘›)๐‘ฅ=๐‘‡(๐‘›โˆ’๐‘ก)๐‘‡(๐‘ก)๐‘ฅ=0 for every ๐‘›โˆˆโ„•, ๐‘›โ‰ฅ๐‘ก and thus (๐‘‡(๐‘›)๐‘ฅ)๐‘›โˆˆโ„•โˆˆโ„ฑ(๐‘‹). It follows that ๐‘ฅโˆˆ๐‘‹1,โ„ฑ, which is not possible since ๐‘ฅโˆˆ๐‘‹2,โ„ฑโงต{0}. Thus, ๐‘‡(๐‘ก)|๐‘‹2,โ„ฑ is one-to-one, for all ๐‘กโ‰ฅ0.

Proposition 4.2. If the pair (โ„ฐ,โ„ฑ) is admissible to the semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then for each ๐‘“โˆˆโ„ฐ(๐‘‹), there exists unique ๐‘ฅ๐‘“โˆˆ๐‘‹2,โ„ฑ such that ๐‘๐‘“(โ‹…;๐‘ฅ๐‘“)โˆˆโ„ฑ(๐‘‹).

Proof. Let ๐‘“โˆˆโ„ฐ(๐‘‹) and ๐‘ฅโˆˆ๐‘‹ from Definition 3.11. For ๐‘ฆ=๐‘ฅโˆ’๐‘ƒ1๐‘ฅ=๐‘ƒ2๐‘ฅ we have that ๐‘ฆโˆˆ๐‘‹2,โ„ฑ and ๐‘๐‘“(๐‘›;๐‘ฆ)=๐‘๐‘“(๐‘›;๐‘ฅ)โˆ’๐‘‡(๐‘›)๐‘ƒ1๐‘ฅ. Since ๐‘๐‘“(โ‹…;๐‘ฅ)โˆˆโ„ฑ(๐‘‹) and (๐‘‡(๐‘›)๐‘ƒ1๐‘ฅ)โˆˆโ„ฑ(๐‘‹), it follows that ๐‘๐‘“(โ‹…;๐‘ฆ)โˆˆโ„ฑ(๐‘‹).
To prove the uniqueness of ๐‘ฆ, suppose that there exists ๐‘งโˆˆ๐‘‹2,โ„ฑ with the property ๐‘๐‘“(โ‹…;๐‘ง)โˆˆโ„ฑ(๐‘‹). Since ๐‘๐‘“(๐‘›;๐‘ฆ)โˆ’๐‘๐‘“(๐‘›;๐‘ง)=๐‘‡(๐‘›)(๐‘ฆโˆ’๐‘ง), we have that ๐‘ฆโˆ’๐‘งโˆˆ๐‘‹1,โ„ฑโˆฉ๐‘‹2,โ„ฑ and therefore ๐‘ง=๐‘ฆ.
The unique vector ๐‘ฆโˆˆ๐‘‹2,โ„ฑ will be denoted by ๐‘ฅ๐‘“.

Proposition 4.3. If the pair (โ„ฐ,โ„ฑ) is admissible to the semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, there exists ๐พ>0 such that โ€–โ€–๐‘ฅ๐‘“โ€–โ€–โ‰ค๐พโ€–๐‘“โ€–โ„ฐ(๐‘‹),โ€–โ€–๐‘๐‘“๎€ทโ‹…;๐‘ฅ๐‘“๎€ธโ€–โ€–โ„ฑ(๐‘‹)โ‰ค๐พโ€–๐‘“โ€–โ„ฐ(๐‘‹),(4.3) for all ๐‘“โˆˆโ„ฐ(๐‘‹).

Proof. We define the operator ๐’ฐโˆถโ„ฐ(๐‘‹)โŸถ๐‘‹2,โ„ฑ๎€ท๐‘ฅโŠ•โ„ฑ(๐‘‹),๐’ฐ๐‘“=๐‘“,๐‘๐‘“๎€ทโ‹…;๐‘ฅ๐‘“.๎€ธ๎€ธ(4.4) It is obvious that ๐’ฐ is a linear operator. Now, we will show that it is also closed.
Let (๐‘“๐‘›)๐‘›โˆˆโ„•โŠ‚โ„ฐ(๐‘‹) such that โ€–๐‘“๐‘›โˆ’๐‘“โ€–โ„ฐ(๐‘‹)โˆ’โˆ’โ†’๐‘›โ†’โˆž0 and โ€–โ€–๐’ฐ๐‘“๐‘›โ€–โ€–โˆ’(๐‘ฆ,๐‘”)๐‘‹2,โ„ฑโŠ•โ„ฑ(๐‘‹)โˆ’โˆ’โˆ’โ†’๐‘›โ†’โˆž0,(4.5) where ๐‘“โˆˆโ„ฐ(๐‘‹), ๐‘ฆโˆˆ๐‘‹2,โ„ฑ, and ๐‘”โˆˆโ„ฑ(๐‘‹). For each ๐‘›โˆˆโ„•, we take ๐‘ฅ๐‘›=๐‘ฅ๐‘“๐‘›โˆˆ๐‘‹2,โ„ฑ and ๐‘ข๐‘›=๐‘๐‘“๐‘›(โ‹…;๐‘ฅ๐‘›)โˆˆโ„ฑ(๐‘‹). We have that โ€–๐‘ฅ๐‘›โˆ’๐‘ฆโ€–โˆ’โˆ’โ†’๐‘›โ†’โˆž0 and โ€–๐‘ข๐‘›โˆ’๐‘”โ€–โ„ฑ(๐‘‹)โˆ’โˆ’โ†’๐‘›โ†’โˆž0.
Since โ€–โ€–๐‘ข๐‘›โ€–โ€–โ‰ค1(๐‘š)โˆ’๐‘”(๐‘š)๐›ฝ๐นโ€–โ€–๐‘ข(0)๐‘›โ€–โ€–โˆ’๐‘”โ„ฑ(๐‘‹),(4.6) we have that lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘š)=๐‘”(๐‘˜), for all ๐‘šโˆˆโ„•. On the other hand, โ€–โ€–๐‘ข๐‘›(๐‘š)โˆ’๐‘๐‘“โ€–โ€–=โ€–โ€–โ€–โ€–๎€ท๐‘ฅ(๐‘š;๐‘ฆ)๐‘‡(๐‘š)๐‘›๎€ธ+โˆ’๐‘ฆ๐‘š๎“๐‘—=0๎€ท๐‘“๐‘‡(๐‘šโˆ’๐‘—)๐‘›๎€ธโ€–โ€–โ€–โ€–โ€–โ€–๐‘ฅ(๐‘—)โˆ’๐‘“(๐‘—)โ‰คโ€–๐‘‡(๐‘š)โ€–๐‘›โ€–โ€–+โˆ’๐‘ฆ๐‘š๎“๐‘—=0โ€–โ€–๐‘“โ€–๐‘‡(๐‘šโˆ’๐‘—)โ€–๐‘›โ€–โ€–,(๐‘—)โˆ’๐‘“(๐‘—)(4.7) which implies lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘š)=๐‘๐‘“(๐‘š;๐‘ฆ), for all ๐‘šโˆˆโ„•. It follows that ๐‘๐‘“(โ‹…;๐‘ฆ)=๐‘”โˆˆโ„ฑ(๐‘‹) and by Proposition 4.2 we have ๐‘ฆ=๐‘ฅ๐‘“. Therefore ๐’ฐ๐‘“=(๐‘ฆ;๐‘”).
Hence, ๐’ฐ is a closed linear operator, and by the Closed-Graph Theorem it is also bounded which means that there exists ๐พ>0 such that โ€–โ€–๐‘ฅ๐‘“โ€–โ€–+โ€–โ€–๐‘๐‘“๎€ทโ‹…;๐‘ฅ๐‘“๎€ธโ€–โ€–โ„ฑ(๐‘‹)=โ€–๐’ฐ๐‘“โ€–๐‘‹2,โ„ฑโŠ•โ„ฑ(๐‘‹)โ‰ค๐พโ€–๐‘“โ€–โ„ฐ(๐‘‹),(4.8) 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 (โ„ฐ,โ„ฑ) is admissible to the semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then (โ„ฐ,โ„“โˆž) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 and โ€–๐‘๐‘“(๐‘›;๐‘ฅ๐‘“)โ€–โ‰ค(๐พ/๐›ฝโ„ฑ(0))โ€–๐‘“โ€–โ„ฐ(๐‘‹), for all ๐‘“โˆˆโ„ฐ(๐‘‹) and ๐‘›โˆˆโ„•.
With this intermediate result we are able to prove that the admissibility of (โ„ฐ,โ„ฑ) to an exponentially bounded semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is a sufficient condition for a no past exponential dichotomy of {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0. The restriction over such a pair (โ„ฐ,โ„ฑ) is that โ„ฐ and โ„ฑ are not simultaneously the bounds of the chain of sequence Schรคffer spaces (in the sense of Proposition 2.8).

Theorem 4.5. Let โ„ฐ and โ„ฑ be sequence Schรคffer spaces such that โ„“1โ‰ โ„ฐ or โ„“โˆž0โŠ„โ„ฑ and let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If (โ„ฐ,โ„ฑ) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then it has a no past exponential dichotomy and ๐‘‹1,โ„ฑ=๐‘‹1.

Proof (Part I). The exponential decay of {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 on ๐‘‹1,โ„ฑ.
Let ๐‘ฅโˆˆ๐‘‹1,โ„ฑ and consider the sequence ๐‘“โˆถโ„•โŸถ๐‘‹,๐‘“(๐‘›)=๐›ฟ0(๐‘›)๐‘ฅ,(4.9) which is in โ„ฐ(๐‘‹) with โ€–๐‘“โ€–โ„ฐ(๐‘‹)=๐›ฝโ„ฐ(0)โ€–๐‘ฅโ€–. Observe that ๐‘๐‘“(๐‘›;0)=๐‘‡(๐‘›)๐‘ฅ, for all ๐‘›โˆˆโ„•. Then, ๐‘๐‘“(โ‹…;0)โˆˆโ„ฑ(๐‘‹) and from โ€–๐‘๐‘“(๐‘›;0)โ€–โ‰ค๐พ/(๐›ฝ๐น(0))โ€–๐‘“โ€–โ„ฐ(๐‘‹) we obtain โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ค๐พ(๐›ฝโ„ฐ(0)/๐›ฝ๐น(0))โ€–๐‘ฅโ€–, for all ๐‘›โˆˆโ„•. For ๐‘กโ‰ฅ0, taking ๐‘›=[๐‘ก] we have โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘€๐‘’๐œ”(๐‘กโˆ’๐‘›)โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ค๐‘€๐‘’๐œ”๐พ๐›ฝโ„ฐ(0)๐›ฝ๐นโ€–(0)๐‘ฅโ€–.(4.10) Since the constant ๐ถ1โˆถ=๐‘€๐‘’๐œ”๐พ(๐›ฝโ„ฐ(0)/๐›ฝ๐น(0)) does not depend on ๐‘ฅ, we can write down โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐ถ1โ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹1,โ„ฑ.(4.11) For ๐‘›,๐‘šโˆˆโ„• and ๐‘ฅโˆˆ๐‘‹1,โ„ฑ, we evaluate โ€–๐‘‡(๐‘›)๐‘ฅโ€–n๎“๐‘—=0๐›ฟ๐‘—(๐‘š)=๐‘›๎“๐‘—=0โ€–๐‘‡(๐‘›โˆ’๐‘—)๐‘‡(๐‘—)๐‘ฅโ€–๐›ฟ๐‘—(๐‘š)โ‰ค๐ถ1๐‘›๎“๐‘—=0โ€–๐‘‡(๐‘—)๐‘ฅโ€–๐›ฟ๐‘—(๐‘š)โ‰ค๐ถ1โˆž๎“๐‘—=0โ€–๐‘‡(๐‘—)๐‘ฅโ€–๐›ฟ๐‘—(๐‘š)=๐ถ1โ€–โ€–๐‘๐‘“โ€–โ€–,(๐‘š;0)(4.12) which implies โˆ‘โ€–๐‘‡(๐‘›)๐‘ฅโ€–๐‘›๐‘—=0๐›ฟ๐‘—โ‰ค๐ถ1โ€–๐‘๐‘“(โ‹…;0)โ€– and therefore โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ€–๐œ’{0,1,โ€ฆ,๐‘›}โ€–โ„ฑโ‰ค๐ถ1โ€–๐‘๐‘“(โ‹…;0)โ€–โ„ฑ(๐‘‹).(4.13) By Proposition 4.3 we get that โ€–๐ถ๐‘‡(๐‘›)๐‘ฅโ€–โ‰ค1๐พ๐›ฝโ„ฐ(0)๐›ฝโ„ฑโ€–(๐‘›)๐‘ฅโ€–,โˆ€๐‘›โˆˆโ„•,๐‘ฅโˆˆ๐‘‹1,โ„ฑ.(4.14)
If โ„“โˆž0โŠ„โ„ฑ, then ๐›ฝโ„ฑ is not bounded and therefore there exists ๐‘›0โˆˆโ„•โˆ— such that ๐œ‚โˆถ=๐ถ1๐พ๐›ฝโ„ฐ(0)/๐›ฝโ„ฑ(๐‘›0)<1 and โ€–๐‘‡(๐‘›0)๐‘ฅโ€–โ‰ค๐œ‚โ€–๐‘ฅโ€–, for all ๐‘ฅโˆˆ๐‘‹1,โ„ฑ.
If โ„“โˆž0โŠ‚โ„ฑ, then โ„ฐโ‰ โ„“1. From Proposition 2.8, it follows that there exists โ„Žโˆˆโ„ฐโงตโ„“1. Consider ๐›พโˆถโ„•โŸถโ„,๐›พ(๐‘›)=๐‘›๎“๐‘—=0||||,โ„Ž(๐‘—)(4.15) which is nondecreasing and lim๐‘›โ†’โˆž๐›พ(๐‘›)=โˆž. For ๐‘›โˆˆโ„• and ๐‘ฅโˆˆ๐‘‹1,โ„ฑ, the sequence ๐‘”โˆถโ„•โŸถ๐‘‹,๐‘”(๐‘š)=๐œ’{0,1,โ€ฆ,๐‘›}||||(๐‘š)โ„Ž(๐‘š)๐‘‡(๐‘š)๐‘ฅ(4.16) has finite support, and thus ๐‘”โˆˆโ„ฐ(๐‘‹). Observing that โ€–๐‘”(๐‘š)โ€–โ‰ค๐ถ1โ€–๐‘ฅโ€–|โ„Ž(๐‘š)| for every ๐‘šโˆˆโ„•, we are led to the evaluation โ€–๐‘”โ€–โ„ฐ(๐‘‹)โ‰ค๐ฟโ€–โ„Žโ€–โ„ฐโ€–๐‘ฅโ€–. Also, ๐‘๐‘”(๐‘š;0)=๐‘š๎“๐‘—=0๐‘‡(๐‘šโˆ’๐‘—)๐‘”(๐‘—)=๐‘š๎“๐‘—=0||||๐œ’๐‘‡(mโˆ’๐‘—)๐‘‡(๐‘—)๐‘ฅโ„Ž(๐‘—){0,1,โ€ฆ,๐‘›}=๎ƒฉ(๐‘—)๐‘š๎“๐‘—=0๐œ’{0,1,โ€ฆ,๐‘›}||โ„Ž||๎ƒช๐‘‡(๐‘—)(๐‘—)(๐‘š)๐‘ฅ,(4.17) and thus we have โ€–๐‘๐‘”(๐‘š;0)โ€–โ‰ค๐›พ(๐‘›)โ€–๐‘‡(๐‘š)๐‘ฅโ€–, for all ๐‘šโˆˆโ„•. Since ๐‘ฅโˆˆ๐‘‹1,โ„ฑ, (๐‘‡(๐‘š)๐‘ฅ)๐‘šโˆˆโ„•โˆˆโ„ฑ(๐‘‹), we have that ๐‘๐‘”(โ‹…;0)โˆˆโ„ฑ(๐‘‹). Taking ๐‘š=๐‘› in โ€–๐‘๐‘”(๐‘š;0)โ€–โ‰ค(๐พ/๐›ฝโ„ฑ(0))โ€–๐‘”โ€–โ„ฐ(๐‘‹), we can deduce that ๐›พ(๐‘›)โ€–๐‘‡(๐‘›)๐‘ฅโ€–โ‰ค๐พ๐ถ1๐›ฝโ„ฑโ€–(0)โ„Žโ€–โ„ฐโ€–๐‘ฅโ€–.(4.18) It follows that there exists ๐‘›0โˆˆโ„•โˆ— such that ๐œ‚โˆถ=(๐พ๐ถ1โ€–โ„Žโ€–โ„ฐ/๐›ฝโ„ฑ(0)๐›พ(๐‘›0))<1 and โ€–๐‘‡(๐‘›0)๐‘ฅโ€–โ‰ค๐œ‚โ€–๐‘ฅโ€–, for all ๐‘ฅโˆˆ๐‘‹1,โ„ฑ.
In both cases, we obtained the existence of some ๐‘›0โˆˆโ„•โˆ— and some constant ๐œ‚โˆˆ(0,1) such that โ€–๐‘‡(๐‘›0)๐‘ฅโ€–โ‰ค๐œ‚โ€–๐‘ฅโ€– for all ๐‘ฅโˆˆ๐‘‹1,โ„ฑ. Therefore, the semigroup {๐‘‡(๐‘ก)|๐‘‹1,โ„ฑ}๐‘กโ‰ฅ0 is exponentially stable (see Remark 3.7). Subsequently, there exist ๐‘1,๐œˆ1>0 such that โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘1๐‘’โˆ’๐œˆ1๐‘กโ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹1,โ„ฑ.(4.19)

Proof (Part II). The exponential blow-up of {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 on ๐‘‹2,โ„ฑ.
Let ๐‘›โˆˆโ„•, ๐‘›0โˆˆโ„•โˆ— and ๐‘ฅโˆˆ๐‘‹2,โ„ฑโงต{0}, and consider the sequence ๐‘“โˆถโ„•โŸถ๐‘‹,๐‘“(๐‘š)=๐›ฟ๐‘›+๐‘›0(๐‘š)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–.(4.20) We have that ๐‘“โˆˆโ„ฐ(๐‘‹) with โ€–๐‘“โ€–โ„ฐ(๐‘‹)=๐›ฝโ„ฐ(0). On the one hand, โˆ’โˆž๎“๐‘—=๐‘š+1๐›ฟ๐‘›+๐‘›0๐‘‡(๐‘—)(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–=โˆ’โˆž๎“๐‘—=0๐›ฟ๐‘›+๐‘›0๐‘‡(๐‘—)(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–+๐‘š๎“๐‘—=0๐›ฟ๐‘›+๐‘›0(๐‘˜)๐‘‡(๐‘šโˆ’๐‘—)๐‘‡(๐‘—)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๎ƒฉ=๐‘‡(๐‘š)โˆ’๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๎ƒช+๐‘š๎“๐‘—=0๐‘‡(๐‘šโˆ’๐‘—)๐‘“(๐‘—)=๐‘๐‘“(๐‘š;๐‘ฆ),(4.21) for all ๐‘šโˆˆโ„• (where ๐‘ฆ=โˆ’๐‘ฅ/โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–โˆˆ๐‘‹2,โ„ฑ), while on the other hand โˆ’โˆž๎“๐‘—=๐‘š+1๐›ฟ๐‘›+๐‘›0(๐‘—)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–=โŽงโŽชโŽจโŽชโŽฉโˆ’๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–,๐‘š<๐‘›+๐‘›0,0,๐‘šโ‰ฅ๐‘›+๐‘›0,(4.22) which implies that ๐‘๐‘“(โ‹…;๐‘ฆ)โˆˆโ„ฑ(๐‘‹). Then, โ€–๐‘๐‘“(๐‘š;๐‘ฆ)โ€–โ‰ค๐พ/(๐›ฝโ„ฑ(0))โ€–๐‘“โ€–โ„ฐ(๐‘‹) and โ€–๐›ฝ๐‘‡(๐‘š)๐‘ฅโ€–โ‰ค๐พโ„ฐ(0)๐›ฝโ„ฑโ€–โ€–๐‘‡๎€ท(0)๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–,โˆ€0โ‰ค๐‘š<๐‘›+๐‘›0.(4.23) Taking now ๐‘š=๐‘›=0 we can write down โ€–โ€–๐‘‡๎€ท๐‘›0๎€ธ๐‘ฅโ€–โ€–โ‰ฅ๐พโˆ’1๐›ฝโ„ฑ(0)๐›ฝโ„ฐ(0)โ€–๐‘ฅโ€–,โˆ€๐‘›0โˆˆโ„•โˆ—,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.24) For ๐‘กโ‰ฅ0, we put ๐‘›0=[๐‘ก]+1 to evaluate ๐‘€๐‘’๐œ”โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅโ€–๐‘‡(๐‘›0)๐‘ฅโ€–โ‰ฅ๐พโˆ’1(๐›ฝโ„ฑ(0)/๐›ฝโ„ฐ(0))โ€–๐‘ฅโ€–. Denoting ๐ถ2โˆถ=(๐พ๐‘€๐‘’๐œ”)โˆ’1๐›ฝโ„ฑ(0)/๐›ฝโ„ฐ(0) we have that โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐ถ2โ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.25) Also, from (4.23), taking ๐ถ๎…žโˆถ=1+๐พ(๐›ฝโ„ฐ(0)/๐›ฝโ„ฑ(0)) we have that ๐ถ๎…žโ€–๐‘‡(โ„Ž)๐‘ฅโ€–โ‰ฅโ€–๐‘‡(๐‘›)๐‘ฅโ€–,โˆ€โ„Žโ‰ฅ๐‘›,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.26) Since ๐‘›,๐‘›0, and ๐‘ฅ are randomly taken (see the beginning of Part II), we have that (โ€–๐‘‡๐‘›)๐‘ฅโ€–โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๐‘›+๐‘›0โˆ’1๎“๐‘—=๐‘›๐›ฟ๐‘—(๐‘š)โ‰ค๐ถโ€ฒ๐‘›+๐‘›0โˆ’1๎“๐‘—=๐‘›(โ€–๐‘‡๐‘—)๐‘ฅโ€–โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๐›ฟ๐‘—(๐‘š)=๐ถโ€ฒ๐‘›+๐‘›0โˆ’1๎“๐‘—=๐‘›โ€–โ€–๐‘๐‘“โ€–โ€–๐›ฟ(๐‘—;๐‘ฆ)๐‘—โ€–โ€–๐‘(๐‘š)โ‰ค๐ถโ€ฒ๐‘“โ€–โ€–(๐‘š;๐‘ฆ),โˆ€๐‘šโˆˆโ„•,(4.27) and therefore โ€–๐‘‡(๐‘›)๐‘ฅโ€–/โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–๐œ’{๐‘›,๐‘›+1,โ€ฆ,๐‘›+๐‘›0โˆ’1}โ‰ค๐ถโ€ฒโ€–๐‘๐‘“(โ‹…;๐‘ฆ)โ€–. It follows that (โ€–๐‘‡๐‘›)๐‘ฅโ€–โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–โ€–โ€–๐‘…๐‘›๐œ’{0,1,โ€ฆ,๐‘›0โˆ’1}โ€–โ€–โ„ฑโ€–โ€–โ€–โ€–๐‘โ‰ค๐ถโ€ฒ๐‘“โ€–โ€–โ€–โ€–(โ‹…;๐‘ฆ)โ„ฑ,(4.28) or equivalently, (โ€–๐‘‡๐‘›)๐‘ฅโ€–โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๐›ฝโ„ฑ๎€ท๐‘›0๎€ธโ€–โ€–๐‘โˆ’1โ‰ค๐ถโ€ฒ๐‘“โ€–โ€–(โ‹…;๐‘ฆ)โ„ฑ(๐‘‹).(4.29) Using the fact that (4.29)โ€–๐‘๐‘“(โ‹…;๐‘ฆ)โ€–โ„ฑ(๐‘‹)โ‰ค๐พโ€–๐‘“โ€–โ„ฐ(๐‘‹), we deduce that โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–โ‰ฅ๐›ฝโ„ฑ๎€ท๐‘›0๎€ธโˆ’1๐ถโ€ฒ๐พ๐›ฝโ„ฐ(0)โ€–๐‘‡(๐‘›)๐‘ฅโ€–,โˆ€๐‘›โˆˆโ„•,๐‘›0โˆˆโ„•โˆ—,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.30)
If โ„“โˆž0โŠ„โ„ฑ, then ๐›ฝโ„ฑ is not bounded and therefore there exists ๐‘›0โˆˆโ„•โˆ— such that ๐œ‚โˆถ=(๐›ฝโ„ฑ(๐‘›0โˆ’1))/(๐ถ๎…ž๐พ๐›ฝโ„ฐ(0))>1 and โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–โ‰ฅ๐œ‚โ€–๐‘‡(๐‘›)๐‘ฅโ€– for all ๐‘›โˆˆโ„• and ๐‘ฅโˆˆ๐‘‹2,โ„ฑ.
If โ„“โˆž0โŠ‚โ„ฑ, then โ„ฐโ‰ โ„“1 and therefore there exists โ„Žโˆˆโ„ฐโงตโ„“1. Consider ๐›พ as in (4.15), and for ๐‘›,๐‘›0โˆˆโ„• and ๐‘ฅโˆˆ๐‘‹2,โ„ฑโงต{0} we define ๐‘”โˆถโ„•โŸถ๐‘‹,๐‘”(๐‘š)=๐‘›0๎“๐‘—=0๐›ฟ๐‘›+๐‘—||||(๐‘š)โ„Ž(๐‘—)๐‘‡(๐‘›+๐‘—)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–.(4.31) Since โ€–๐‘”(๐‘š)โ€–โ‰ค๐ถโ€ฒ๐œ’{๐‘›,๐‘›+1,โ€ฆ,๐‘›+๐‘›0}(๐‘š)|(๐‘…๐‘›โ„Ž)(๐‘š)|โ‰ค๐ถโ€ฒ|(๐‘…๐‘›โ„Ž)(๐‘š)| for each ๐‘šโˆˆโ„•, we have that ๐‘”โˆˆโ„ฐ(๐‘‹) with โ€–๐‘”โ€–โ„ฐ(๐‘‹)โ‰ค๐ถโ€ฒโ€–๐‘…๐‘›โ„Žโ€–โ„ฐ=๐ถโ€ฒโ€–โ„Žโ€–โ„ฐ. Note that ๐‘”(๐‘š)=๐œ’{๐‘›,๐‘›+1,โ€ฆ,๐‘›+๐‘›0}(๐‘š)|๐‘…๐‘›โ„Ž(๐‘š)|๐‘‡(๐‘š)๐‘ฅ/โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–, for all ๐‘šโˆˆโ„•. On the one hand, โˆ’โˆž๎“๐‘—=๐‘š+1๐œ’{๐‘›,โ€ฆ,๐‘›+๐‘›0}(||๐‘…๐‘—)๐‘›||โ„Ž(๐‘—)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–=โˆ’โˆž๎“๐‘—=0๐œ’{๐‘›,โ€ฆ,๐‘›+๐‘›0}(||๐‘…๐‘—)๐‘›||โ„Ž(๐‘—)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–++๐‘š๎“๐‘—=0๐‘‡(๐‘šโˆ’๐‘—)๐œ’{๐‘›,โ€ฆ,๐‘›+๐‘›0}||๐‘…(๐‘—)๐‘›||โ„Ž(๐‘—)๐‘‡(๐‘—)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๎ƒฉ๐›พ๎€ท๐‘›=๐‘‡(๐‘š)0๎€ธโˆ’๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–๎ƒช+๐‘š๎“๐‘—=0๐‘‡(๐‘šโˆ’๐‘—)๐‘”(๐‘—)=๐‘๐‘”(๐‘š;๐‘ง),(4.32) for all ๐‘šโˆˆโ„• (where ๐‘งโˆถ=๐›พ(๐‘›0)(โˆ’๐‘ฅ/โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–)โˆˆ๐‘‹2,โ„ฑ), while on the other hand โˆ’โˆž๎“๐‘—=๐‘š+1๐œ’{๐‘›,โ€ฆ,๐‘›+๐‘›0}(||๐‘…๐‘—)๐‘›||โ„Ž(๐‘—)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€ท๐‘›โˆ’๐›พ0๎€ธ๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–โˆ’๎€ท๐›พ๎€ท๐‘›,๐‘š<๐‘›,0๎€ธ๎€ธโˆ’๐›พ(๐‘šโˆ’๐‘›)๐‘‡(๐‘š)๐‘ฅโ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–,๐‘›โ‰ค๐‘š<๐‘›+๐‘›0,0,๐‘šโ‰ฅ๐‘›+๐‘›0.(4.33) From (4.32) and (4.33), it follows that ๐‘๐‘”(โ‹…;๐‘ง) has finite support, and thus it belongs to โ„ฑ(๐‘‹). Therefore, โ€–๐‘๐‘”(๐‘›;๐‘ง)โ€–โ‰ค(๐พ/๐›ฝโ„ฑ(0))โ€–gโ€–โ„ฐ(๐‘‹) and using that โ€–๐‘”โ€–โ„ฐ(๐‘‹)โ‰ค๐ถ๎…žโ€–โ„Žโ€–โ„ฐ we obtain that ๎€ท๐›พ๎€ท๐‘›0๎€ธโˆ’||||๎€ธโ„Ž(0)โ€–๐‘‡(๐‘›)๐‘ฅโ€–๎€ทโ€–๐‘‡๐‘›+๐‘›0๎€ธโ‰ค๐‘ฅโ€–๐พ๐ถโ€ฒ๐›ฝโ„ฑ(0)โ€–โ„Žโ€–โ„ฐ.(4.34) Thus, โ€–๐‘‡(๐‘›+๐‘›0)๐‘ฅโ€–โ‰ฅ(๐›ฝโ„ฑ(0)/๐พ๐ถ๎…žโ€–โ„Žโ€–โ„ฐ)(๐›พ(๐‘›0)โˆ’|โ„Ž(0)|)โ€–๐‘‡(๐‘›)๐‘ฅโ€–, for all ๐‘›,๐‘›0โˆˆโ„• and ๐‘ฅโˆˆ๐‘‹2,โ„ฑ. Then, there exist ๐‘›0โˆˆโ„•โˆ— and ๐œ‚>1 such that โ€–โ€–๐‘‡๎€ท๐‘›+๐‘›0๎€ธ๐‘ฅโ€–โ€–โ‰ฅ๐œ‚โ€–๐‘‡(๐‘›)๐‘ฅโ€–,โˆ€๐‘›โˆˆโ„•,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.35)
In both cases, we obtained the existence of some ๐‘›0โˆˆโ„•โˆ— and some constant ๐œ‚>1 satisfying the condition (ii) from Lemma 3.9, while the first condition is assured by (4.25). Subsequently, there exist ๐‘2,๐œˆ2>0 such that โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ฅ๐‘2๐‘’๐œˆ2๐‘กโ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.36)
If we take ๐œˆโˆถ=min{๐œˆ1,๐œˆ2}>0, all the conditions guaranteeing the existence of a no past exponential dichotomy are met.

Proof (Part III). We prove that ๐‘‹1,โ„ฑ=๐‘‹1, no matter how we choose the sequence Schรคffer space โ„ฑ. If ๐‘ฅโˆˆ๐‘‹1,โ„ฑ, then โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐‘1๐‘’โˆ’๐œˆ1๐‘กโ€–๐‘ฅโ€–, for all ๐‘กโ‰ฅ0, which implies ๐‘ฅโˆˆ๐‘‹1.
Conversely, let ๐‘ฅโˆˆ๐‘‹1, ๐‘ขโˆˆ๐‘‹1,โ„ฑ, and ๐‘ฃโˆˆ๐‘‹2,โ„ฑ such that ๐‘ฅ=๐‘ข+๐‘ฃ. For every ๐‘›โˆˆโ„•, we have that โ€–๐‘‡(๐‘›)๐‘ฅโ€–=โ€–๐‘‡(๐‘›)๐‘ฃโˆ’(โˆ’๐‘‡(๐‘›)๐‘ข)โ€–โ‰ฅโ€–๐‘‡(๐‘›)๐‘ฃโ€–โˆ’โ€–๐‘‡(๐‘›)๐‘ขโ€–โ‰ฅ๐‘2๐‘’๐œˆ2๐‘›โ€–๐‘ฃโ€–โˆ’๐‘1๐‘’โˆ’๐œˆ1๐‘›โ€–๐‘ขโ€–.(4.37) If we suppose that ๐‘ฃโ‰ 0, then โ€–๐‘‡(๐‘›)๐‘ฅโ€–โˆ’โˆ’โ†’๐‘›โ†’โˆžโˆž contradicting the fact that ๐‘ฅโˆˆ๐‘‹1. It follows immediately that ๐‘ฃ=0 and ๐‘ฅ=๐‘ขโˆˆ๐‘‹1,โ„ฑ.

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 (โ„“๐‘,โ„“๐‘ž)1โ‰ค๐‘โ‰ค๐‘žโ‰คโˆž and (๐‘,๐‘ž)โ‰ (1,โˆž) and the hyperbolicity of a ๐ถ0-semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, when we assume a priori that the kernel of the splitting projection is ๐‘‡(๐‘ก)-invariant and ๐‘‡(๐‘ก)|Ker๐‘ƒ 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 ๐‘‡(๐‘ก)|Ker๐‘ƒ is invertible (we do not even assume that Ker๐‘ƒ is ๐‘‡(๐‘ก)-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 โ€œoutputโ€ or โ€œnice discrete-time mild solutionsโ€ (i.e., the solution of the inhomogeneous difference equation problem) to stay in โ„“๐‘ž(๐‘‹), as before. Moreover, this approach can provide interesting input spaces (i.e., the spaces consisting in "test sequences") which are different from the classical โ„“๐‘ 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 (โ„ฐ,โ„ฑ), which is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, and with the property that โ„“1โ‰ โ„ฐ or โ„“โˆž0โŠ„โ„ฑ, then the subspace ๐‘‹1,โ„ฑ (which induces the no past exponential dichotomy) is actually the regular stable subspace ๐‘‹1. If we would impose in addition that the complement of ๐‘‹1 (denoted by ๐‘‹2) is also ๐‘‡(๐‘ก)-invariant, then the aforementioned admissibility condition would imply that {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 extends automatically to a ๐ถ0-group on ๐‘‹2, and thus we would get hyperbolicity for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 (see Theorem 4.12 below). Therefore, we can conclude that โ€œadmissibilityโ€ converts to โ€œno past exponential dichotomyโ€โ€œadmissibilityโ€ and โ€œ๐‘‡(๐‘ก)-invariance of ๐‘‹2โ€ converts to โ€œhyperbolicityโ€.

The next example shows that the condition โ€œ๐ธโ‰ โ„“1 or โ„“โˆž0โŠ„โ„ฑโ€ in the statement of Theorem 4.5 is essential.

Example 4.7. Let ๐‘‹=โ„ and consider the semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 by ๐‘‡(๐‘ก)=๐ผโ„ for all ๐‘กโ‰ฅ0. If ๐‘“โˆˆโ„“1, then there exists โˆ‘๐‘ฅ=โˆ’โˆž๐‘˜=0๐‘“(๐‘˜)โˆˆโ„ (the series being absolutely convergent) such that ๐‘๐‘“(โ‹…;๐‘ฅ)โˆˆโ„“โˆž0(๐‘‹). Therefore, the pair (โ„“1,โ„“โˆž0) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, but one can easily check that {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 does not posses a no past exponential dichotomy.
The above theorem shows that the space ๐‘‹1,โ„ฑ induces a no past exponential dichotomy for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0โ€”an exponentially bounded semigroupโ€”and that ๐‘‹1,โ„ฑ is actually ๐‘‹1. Concerning the hypothesis over โ„ฐ and โ„ฑ 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: (๐ต1,๐ท1) is said to be stronger than (๐ต,๐ท) (or the later pair is weaker than the former) if ๐ตโ†ช๐ต1 and ๐ท1โ†ช๐ท. 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 (โ„“1,โ„“โˆž0) is stronger than (โ„ฐ,โ„ฑ) if โ„ฐโŠ‚โ„“1 and โ„“โˆž0โŠ‚โ„ฑ, or equivalently (considering Proposition 2.8), โ„ฐ=โ„“1 and โ„“โˆž0โŠ‚โ„ฑ. Therefore, the hypothesis โ€œ๐ธโ‰ โ„“1 or โ„“โˆž0โŠ„โ„ฑโ€ can be replaced by, โ€œthe pair (โ„“1,โ„“โˆž0) is not stronger than the pair (โ„ฐ,โ„ฑ)โ€.
Dropping off the restriction โ€œโ„“1โ‰ โ„ฐ or โ„“โˆž0โŠ„โ„ฑโ€, 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 {๐‘‡(๐‘ก)|๐‘‹1,โ„ฑ}๐‘กโ‰ฅ0 (in the sense of Definition 3.6).

Corollary 4.9. Let (โ„ฐ,โ„ฑ) be a pair sequence Schรคffer space and, {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If the pair (โ„ฐ,โ„ฑ) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an ordinary dichotomy; that is, there exist two constants ๐ถ1,๐ถ2>0 such that โ€–๐‘‡(๐‘ก)๐‘ฅโ€–โ‰ค๐ถ1โ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹1,โ„ฑ,(โ€–๐‘‡๐‘ก)๐‘ฅโ€–โ‰ฅ๐ถ2โ€–๐‘ฅโ€–,โˆ€๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐‘‹2,โ„ฑ.(4.38)

Proof. Note that in the proof of Theorem 4.5, we do not use the hypothesis โ€œ๐ธโ‰ โ„“1 or โ„“โˆž0โŠ„โ„ฑโ€ to prove the relations (4.11) and (4.25).

Corollary 4.10. Let ๐‘,๐‘žโˆˆ[1,โˆž] such that (๐‘,๐‘ž)โ‰ (1,โˆž) and let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If the pair (โ„“๐‘,โ„“๐‘ž) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has a no past exponential dichotomy.

Proof. We have the equivalence (due to Proposition 2.10 and the fact that ๐›ฝโ„“๐‘ž(๐‘›)=(๐‘›+1)1/๐‘ž for each ๐‘›โˆˆโ„•): โ„“โˆž0โŠ‚โ„“๐‘ž if and only if ๐‘ž=โˆž. Thus, given (๐‘,๐‘ž)โ‰ (1,โˆž) the hypothesis of Theorem 4.5 is satisfied and the conclusion is immediate.

Corollary 4.11. Let โ„“ฮฆ be a sequence Orlicz space and let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If the pair (โ„“ฮฆ,โ„“ฮฆ) is admissible to {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0, then {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 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 ๐‘‡(๐‘ก)-invariance of ๐‘‹2, we can deduce the invertibility of ๐‘‡(๐‘ก)|๐‘‹2, thus obtaining the exponential dichotomy (or equivalently, hyperbolicity) for {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0. Note that the first condition of the following theorem does not require that ๐‘‹2 is invariant to all operators ๐‘‡(๐‘ก). However, we can prove that {๐‘‡(๐‘ก)|๐‘‹2}๐‘กโ‰ฅ0 is a semigroup (exponentially bounded if {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 is exponentially bounded); hence the invertibility of all its operators follows from the invertibility of just one of them.

Theorem 4.12. Let โ„ฐ and โ„ฑ be two sequence Schรคffer spaces such that โ„“1โ‰ โ„ฐ or โ„“โˆž0โŠ„โ„ฑ and let {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 be an exponentially bounded semigroup. If(i)there exists ๐‘ก0>0 such that ๐‘ƒ2๐‘‡(๐‘ก)๐‘ƒ2=๐‘‡(๐‘ก)๐‘ƒ2 for all ๐‘กโˆˆ[0,๐‘ก0], (ii)(โ„ฐ,โ„ฑ) is admissible to ๐‘‡, then ๐‘‡(๐‘ก)โˆถ๐‘‹2โ†’๐‘‹2 is invertible for each ๐‘กโ‰ฅ0. Therefore, {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has an exponential dichotomy.

Proof. Let ๐‘ก>๐‘ก0 and consider ๐‘›โˆˆโ„• and ๐›ฟโˆˆ[0,๐‘ก0) such that ๐‘ก=๐‘›๐‘ก0+๐›ฟ. If ๐‘ฅโˆˆ๐‘‹2, we have that ๐‘‡(๐‘ก)๐‘ฅ=๐‘‡(๐›ฟ)๐‘‡(๐‘ก0)๐‘›๐‘ฅโˆˆ๐‘‹2. Therefore, the condition (i) assures that the operator ๐‘‡(๐‘ก)โˆถ๐‘‹2โ†’๐‘‹2 is well defined (with respect to the range) for each ๐‘กโ‰ฅ0 and from Theorem 4.5 we have that {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 has a no past exponential dichotomy.
Let ๐‘กโ‰ฅ0, ๐‘›0=[๐‘ก]+1, ๐‘ฆโˆˆ๐‘‹2 and consider the sequence ๐‘“โˆถโ„•โŸถ๐‘‹,๐‘“(๐‘›)=โˆ’๐›ฟ๐‘›0๎€ท๐‘›(๐‘›)๐‘‡0๎€ธโˆ’๐‘ก๐‘ฆ,(4.39) which belongs to โ„ฐ(๐‘‹) with โ€–๐‘“โ€–โ„ฐ(๐‘‹)=๐›ฝโ„ฐ(0)โ€–๐‘‡(๐‘›0โˆ’๐‘ก)๐‘ฆโ€–. Then, according to Proposition 4.2, there exists a unique ๐‘ฅโˆˆ๐‘‹2 such that ๐‘๐‘“(โ‹…;๐‘ฅ)โˆˆโ„ฑ(๐‘‹). Since ๐‘๐‘“(๐‘›;๐‘ฅ)=๐‘‡(๐‘›)๐‘ฅโˆ’๐‘›๎“๐‘˜=0๐›ฟ๐‘›0๎€ท๐‘›(๐‘˜)๐‘‡(๐‘›โˆ’๐‘˜)๐‘‡0๎€ธ๐‘ฆ๎€ทโˆ’๐‘ก=๐‘‡(๐‘›)๐‘ฅโˆ’๐‘‡๐‘›โˆ’๐‘›0๎€ธ๐‘‡๎€ท๐‘›0๎€ธโˆ’๐‘ก๐‘ฆ=๐‘‡(๐‘›โˆ’๐‘ก)๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘‡(๐‘›โˆ’๐‘ก)๐‘ฆ=๐‘‡(๐‘›โˆ’๐‘ก)(๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฆ),(4.40) for all ๐‘›โˆˆโ„•, ๐‘›โ‰ฅ๐‘›0, we have that ๐‘‡(๐‘ก)๐‘ฅ,๐‘ฆโˆˆ๐‘‹2 and โ€–โ€–๐‘๐‘“โ€–โ€–(๐‘›;๐‘ฅ)=โ€–๐‘‡(๐‘›โˆ’๐‘ก)(๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฆ)โ€–โ‰ฅ๐‘2๐‘’๐œˆ2(๐‘›โˆ’๐‘ก)โ€–๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘›โ‰ฅ๐‘›0.(4.41) If we assume that ๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘ฆโ‰ 0, then lim๐‘›โ†’โˆžโ€–๐‘๐‘“(๐‘›;๐‘ฅ)โ€–=โˆž that contradicts the fact that ๐‘๐‘“(โ‹…;๐‘ฅ)โˆˆโ„“โˆž(๐‘‹). It follows that ๐‘‡(๐‘ก)๐‘ฅ=๐‘ฆ. We proved that ๐‘‡(๐‘ก)|๐‘‹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 ๐‘‹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. It will be clear (see Example 4.17) that for a semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0