Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

Abstract

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 ℓ^𝑝 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 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 infinite-dimensional 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].

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 Perron’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 “second member” 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 Schäffer [8] and Henry [9]. More recently, we have the papers of Ben-Artzi and Gohberg [10], Pinto [11], and LaSalle [12]. Applications of this “discrete-time theory” 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 (𝐴), 𝑥^(𝑡)=𝐴𝑥(𝑡) (assuming its well-posedness, i.e., 𝐴 generates a 𝐶₀-semigroup {𝑇(𝑡)}_𝑡≥0) consists in the existence of a bounded projection 𝑃 such that the solutions that start in 𝑋₁=Im𝑃 decay to zero and the solutions that start in 𝑋₂=Im(𝐼−𝑃) are unbounded. In the hypothesis that 𝑋₂ is 𝑇(𝑡)-invariant and finite-dimensional, the existence of a dichotomy for (𝐴) implies that solutions starting in 𝑋₂ exist in backward time (or equivalently, {𝑇(𝑡)} extends to a 𝐶₀-group on 𝑋₂). 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 𝑇(𝑡)|_𝑋2 are invertible (we do not even assume that 𝑋₂ is 𝑇(𝑡)-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 ℓ^𝑝 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 Schäffer 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 real-valued 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 𝑅𝑓(𝑛)=𝑓(𝑛−1),𝑛∈ℕ^∗0,𝑛=0,,𝐿𝑓(𝑛)=𝑓(𝑛+1),(2.1) 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:(𝑠₁)𝛿₀∈ℰ,(𝑠₂)if 𝑓∈ℰ, then 𝑅𝑓∈ℰ and ‖𝑅𝑓‖_ℰ=‖𝑓‖_ℰ,(𝑠₃)if 𝑓∈𝒮 and 𝑔∈ℰ such that |𝑓|≤|𝑔|, then 𝑓∈ℰ and ‖𝑓‖_ℰ≤‖𝑔‖_ℰ.

Remark 2.2. By (𝑠₁) and (𝑠₂) 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,∞), ℓ^𝑝=𝑓∶ℕ→ℝ∶^∞_𝑘=0||𝑓(𝑘)||^𝑝<∞,withthenorm‖𝑓‖_𝑝=^∞_𝑘=0|𝑓(𝑘)|^𝑝^1/𝑝ℓ_∞,=𝑓∶ℕ→ℝ∶sup_𝑛∈ℕ||𝑓(𝑛)||<∞,withthenorm‖𝑓‖_∞=sup_𝑛∈ℕ||𝑓(𝑛)||.(2.2) The subspace of ℓ^∞, ℓ^∞0={𝑓∈ℓ^∞∶lim_𝑛→∞𝑓(𝑛)=0} (often denoted by 𝑐₀) 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 ℓ¹, ℓ^∞, and ℓ^∞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 𝛼_ℰ(𝑛)=inf𝐿>0∶^𝑛_𝑘=0𝛽||𝑓(𝑘)||≤𝐿‖𝑓‖_ℰ,∀𝑓∈ℰ_ℰ,(𝑛)=‖‖𝜒_{{0,1,…,𝑛}}‖‖_ℰ,(2.3) which are both nondecreasing and 𝛽_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)=∫^𝑡0𝜑(𝑠)𝑑𝑠,𝑡≥0. Consider ℓ^Φ=𝑓∈𝒮∶exists𝑐>0suchthat^𝑛_𝑘=0‖𝑓‖_ΦΦ𝑐||𝑓(𝑘)||<∞withthenorm=inf𝑐>0^𝑛_𝑘=0Φ𝑐⁻¹||𝑓(𝑘)||≤1(theLuxemburgnorm).(2.4) For the Banach space (ℓ^Φ,‖⋅‖_Φ) the conditions (𝑠₁), (𝑠₂), and (𝑠₃) are verified; hence (ℓ^Φ,‖⋅‖_Φ) is a sequence Schäffer space.
For 1≤𝑝<∞, taking 𝜑(𝑡)=𝑝𝑡^𝑝−1 we have that (ℓ^Φ,‖⋅‖_Φ)≡(ℓ^𝑝,‖⋅‖_𝑝). Even ℓ^∞ is a sequence Orlicz space, obtained from 𝜑(𝑡)=0 for 𝑡∈[0,1] and 𝜑(𝑡)=∞ for 𝑡>1 [16, Section  3].

Remark 2.5. Let 𝑝∈[1,∞). If (ℓ^Φ,‖⋅‖_Φ)=(ℓ^𝑝,‖⋅‖_𝑝), then lim_𝑡→0(Φ(𝑡)/𝑡^𝑝)=1.
Indeed, if (ℓ^Φ,‖⋅‖_Φ)=(ℓ^𝑝,‖⋅‖_𝑝), then ‖𝜒_{{0,1,…,𝑛}}‖_Φ=‖𝜒_{{0,1,…,𝑛}}‖_𝑝, for all 𝑛∈ℕ, which is equivalent with Φ⁻¹1𝑛+1=𝑛+11^1/𝑝,∀𝑛∈ℕ.(2.5) Let 𝑥∈(0,1] and 𝑚=[1/𝑥]∈ℕ^∗. Using the fact that Φ⁻¹ is nondecreasing, we have that 𝑚1+1^1/𝑝=Φ⁻¹𝑚1+1≤Φ⁻¹(𝑥)≤Φ⁻¹𝑚1=𝑚1^1/𝑝,(2.6) which implies that ([1/𝑥]1+1)𝑥^1/𝑝≤Φ⁻¹(𝑥)𝑥^1/𝑝≤1𝑥[1/𝑥]^1/𝑝,(2.7) for all 𝑥∈(0,1]. Hence lim_𝑥→0Φ⁻¹(𝑥)𝑥^−(1/𝑝)=1 and lim_𝑢→0Φ(𝑢)𝑢^𝑝=lim_𝑢→01Φ⁻1(Φ(𝑢))/(Φ(𝑢))^1/𝑝^𝑝=1..(2.8)

Example 2.6. Consider 𝜑∶ℝ₊→ℝ₊: 𝜑(𝑡)=^∞_𝑚₌₁^𝑚√𝑡𝑚²,Φ(𝑡)=^𝑡0𝜑(𝑠)𝑑𝑠=^∞_𝑚₌₁𝑡^1+1/𝑚𝑚(𝑚+1).(2.9) We claim that ℓ^Φ≠ℓ^𝑝, no matter how we choose 𝑝∈[1,∞).
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, ℓ^Φ≠ℓ^∞ since 𝜒_ℕ∈ℓ^∞⧵ℓ^Φ.

Remark 2.7. By simple computations we obtain that 𝛼_ℓ^𝑝(𝑛)=(𝑛+1)^1−1/𝑝,𝛽_ℓ^𝑝(𝑛)=(𝑛+1)^1/𝑝(2.10) for 1≤𝑝≤∞ (with the convention 1/∞=0) and for the Orlicz sequence spaces, 𝛼_ℓ^Φ(𝑛)=(𝑛+1)Φ⁻¹1𝑛+1,𝛽^ℓΦ(𝑛)=1Φ⁻¹(1/(𝑛+1)).(2.11)
For two Banach spaces (𝐵₁,‖⋅‖₁) and (𝐵₂,‖⋅‖₂), we say that 𝐵₁ is continuously embedded in 𝐵₂ (𝐵₁↪𝐵₂) if 𝐵₁⊂𝐵₂ and there exists 𝑐>0 such that ‖𝑓‖₂≤𝑐‖𝑓‖₁ for all 𝑓∈𝐵₁. For the following three propositions, proofs can be retrieved from [8, Section  3].

Proposition 2.8. If (ℰ,‖⋅‖_ℰ) is a sequence Schäffer space, then ℓ¹↪ℰ↪ℓ^∞ with 𝛽_ℰ(0)‖𝑓‖_∞≤‖𝑓‖_ℰ for all 𝑓∈ℰ and ‖𝑓‖_ℰ≤𝛽_ℰ(0)‖𝑓‖₁ for all 𝑓∈ℓ¹.

Proposition 2.9. If (ℰ,‖⋅‖_ℰ) and (ℱ,‖⋅‖_ℱ) are sequence Schäffer spaces, then ℰ↪ℱ if and only if ℰ⊂ℱ.

Proposition 2.10. Let (ℰ,‖⋅‖_ℰ) be a sequence Schäffer space. The following characterizations hold:(i)𝛼_ℰ is bounded if and only if ℰ=ℓ¹ and ‖⋅‖₁∼‖⋅‖_ℰ; (ii)𝛽_ℰ is bounded if and only if ℓ^∞0⊂ℰ.

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:(i)if 𝑓∈𝒮(𝑋) has finite support, then 𝑓∈ℰ(𝑋);(ii)if 𝑓∈ℰ(𝑋), then 𝑅𝑓∈ℰ(𝑋) and ‖𝑅𝑓‖_𝐸(𝑋)=‖𝑓‖_ℰ(𝑋);(iii)if 𝑓∈𝒮(𝑋) and 𝑔∈ℰ(𝑋) such that ‖𝑓‖≤‖𝑔‖, then 𝑓∈ℰ(𝑋) and ‖𝑓‖_ℰ(𝑋)≤‖𝑔‖_ℰ(𝑋).

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

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

Let 𝑋 be a Banach space, 𝑥₀∈𝑋, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 a linear operator, and consider the (abstract) Cauchy problem (𝐴;𝑥₀): 𝑥̇(𝑡)=𝑥𝐴(0𝑥)(=𝑡)𝑥,₀𝑡≥0,.(3.1) The function 𝑢∶[0,∞)→𝐷(𝐴) is said to be a classical solution for the previous Cauchy problem if it is continuously differentiable on [0,∞) and satisfies (𝐴;𝑥₀). A theorem of E. Hille states that if the resolvent set of 𝐴 is nonempty, then for every 𝑥∈𝐷(𝐴) the Cauchy problem (𝐴;𝑥) has a unique classical solution if and only if 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup. For the proof of this result we guide the reader to [18, Theorem  6.7, page 150] or [19, Theorem  1.3, page 102].

Definition 3.1. A family of bounded linear operators {𝑇(𝑡)}_𝑡≥0 acting on 𝑋 is called a semigroup if (i)𝑇(0)=𝐼 (where 𝐼 is the identity operator on 𝑋), (ii)𝑇(𝑡+𝑠)=𝑇(𝑡)𝑇(𝑠) for all 𝑡,𝑠≥0.If (ii) holds for any 𝑡,𝑠∈ℝ, then {𝑇(𝑡)}_𝑡≥0 is called a group. If in addition of (i) and (ii), there exist 𝑀,𝜔>0 such that (iii)‖𝑇(t)‖≤𝑀𝑒^𝜔𝑡 for all 𝑡≥0,then {𝑇(𝑡)}_𝑡≥0 is said to be exponentially bounded.

If 𝑌 is a closed subspace of 𝑋 and {𝑇(𝑡)}_𝑡≥0 is a (𝐶₀-)semigroup such that 𝑌 is 𝑇(𝑡)-invariant for any 𝑡≥0, then the restrictions 𝑇(𝑡)|_𝑌 form a (𝐶₀-)semigroup called the subspace (𝐶₀-)semigroup on 𝑌 ([18, page 43]). A (𝐶₀-)semigroup can be extended to a (𝐶₀-)group if and only if there exists 𝑡₀>0 such that 𝑇(𝑡₀) is invertible (see [18, page 80] or [19, Section  1.6]).

The linear operator defined by 𝐷(𝐴)=𝑥∈𝑋∶lim_𝑡↓0𝑇(𝑡)𝑥−𝑥𝐴𝑥=lim_𝑡↓0𝑡exists,𝑇(𝑡)𝑥−𝑥𝑡for𝑥∈𝐷(𝐴)(3.2) is the infinitesimal generator of the semigroup {𝑇(𝑡)}_𝑡≥0. If {𝑇(𝑡)}_𝑡≥0 is a 𝐶₀-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).

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 by𝑃=12𝜋𝑖_Γ(𝜆𝐼−𝑇)⁻¹𝑑𝜆.(3.3) 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 𝜎𝑇||_Im𝑃=𝜎(𝑇)∩𝐷(0,1),𝜎𝑇||_{Im(𝐼−𝑃)}=𝜎(𝑇)∩(ℂ⧵𝐷(0,1)),(3.4) we obtain the spectral radius 𝑟(𝑇|_Im𝑃)<1 and also the operator 𝑇|_{Im(𝐼−𝑃)} is invertible with 𝑟(𝑇⁻¹|_{Im(𝐼−𝑃)})<1. Hence, if 𝑇 is hyperbolic, then there exist the constants 𝑁,𝜈>0 such that, for all integers 𝑛≥0, ‖𝑇‖𝑇^𝑛𝑥‖≤𝑁𝑒^−𝜈𝑛‖𝑥‖,∀𝑥∈Im𝑃,_𝑛𝑥‖≥𝑁⁻¹𝑒^𝜈𝑛‖𝑥‖,∀𝑥∈Im(𝐼−𝑃)(3.5) (for the previous exposure we consulted [14, page 28]).

Definition 3.2. A semigroup {𝑇(𝑡)}_𝑡≥0 is hyperbolic if there exists 𝑡₀>0 such that 𝑇(𝑡₀) 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 𝑃²=𝑃) and the constants 𝑁,𝜈>0 such that (i)𝑃𝑇(𝑡)=𝑇(𝑡)𝑃, for all 𝑡≥0, (ii)𝑇(𝑡)_|Ker𝑃∶Ker𝑃→Ker𝑃 is an isomorphism, for each 𝑡≥0, (iii)‖𝑇(𝑡)𝑥‖≤𝑁𝑒^−𝜈𝑡‖𝑥‖, for all 𝑡≥0 and 𝑥∈Im𝑃, (iv)‖𝑇(𝑡)^−1|Ker𝑃𝑥‖≤𝑁𝑒^−𝜈𝑡‖𝑥‖, for all 𝑡≥0 and 𝑥∈Ker𝑃.

The first condition in the previous definition expresses equivalently that Im𝑃 and Ker𝑃 are both 𝑇(𝑡)-invariant; the essence of (ii) is that {𝑇(𝑡)|_Ker𝑃}_𝑡≥0 can be extended to a group. The next result establishes the relation between hyperbolicity and exponential dichotomy for 𝐶₀-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. (i){𝑇(𝑡)}_𝑡≥0 is hyperbolic. (ii){𝑇(𝑡)}_𝑡≥0 has an exponential dichotomy. Moreover, if (i) holds, then 𝑋=𝑋₁⊕𝑋₂ where 𝑋₁=Im𝑃, 𝑋₂=Im(𝐼−𝑃), and 𝑃=12𝜋𝑖_Γ𝜆𝐼−𝑇𝑡₀⁻¹𝑑𝜆(3.6) is the spectral Riesz projection for 𝑇(𝑡₀) that corresponds to 𝜎(𝑇(𝑡₀))∩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 𝐶₀-semigroup, then the differential equation 𝑥̇=𝐴𝑥 has the property that the solutions 𝑥(⋅) starting from 𝑋₁ (resp., from 𝑋₂) 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 𝑋₂ to exist for negative time, or in counterpart it forces the semigroup to be invertible on 𝑋₂. 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 𝑋₂ with an exponential blow-up in positive time. We will call the “exponential decay on 𝑋₁ and exponential blow-up on 𝑋₂” (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 𝑁₁,𝑁₂,𝜈>0 such that (i)𝑃𝑇(𝑡)𝑃=𝑇(𝑡)𝑃, for all 𝑡≥0, (ii)‖𝑇(𝑡)𝑥‖≤𝑁₁𝑒^−𝜈𝑡‖𝑥‖, for all 𝑡≥0, 𝑥∈Im𝑃, (iii)‖𝑇(𝑡)𝑥‖≥𝑁₂𝑒^𝜈𝑡‖𝑥‖, for all 𝑡≥0, 𝑥∈Ker𝑃.

Definition 3.6. The semigroup {𝑇(𝑡)}_𝑡≥0 has an ordinary dichotomy if there exist a projection 𝑃 and the constants 𝑁₁,𝑁₂>0 such that (i)𝑃𝑇(𝑡)𝑃=𝑇(𝑡)𝑃, for all 𝑡≥0, (ii)‖𝑇(𝑡)𝑥‖≤𝑁₁‖𝑥‖, for all 𝑡≥0, 𝑥∈Im𝑃, (iii)‖𝑇(𝑡)𝑥‖≥𝑁₂‖𝑥‖, for all 𝑡≥0, 𝑥∈Ker𝑃.

Remark 3.7. Note that if 𝑃 is one-to-one, then Im𝑃=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 𝑡₀>0 such that ‖𝑇(𝑡₀)‖<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 𝐶₀-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 𝐿¹(ℝ₊,ℝ), given by 𝑇(𝑡)𝑓(𝑠)=0𝑓,(𝑠−𝑡),𝑠0≥≤𝑡𝑠.<𝑡,(3.7) Then 𝑆(𝑡)=𝑒^𝑡𝐴⊕𝑒^𝑡𝑇(𝑡),𝑆(𝑡)(𝑥,𝑓)=(𝑒^𝑡𝐴𝑥,𝑒^𝑡𝑇(𝑡)𝑓) has a no past exponential dichotomy on 𝑋=ℝ^𝑝⊕𝐿¹(ℝ₊,ℝ) with 𝑋₁={(𝑥,0)∶𝑥∈ℝ^𝑝} and 𝑋₂={(0,𝑓)∶𝑓∈𝐿¹(ℝ₊,ℝ)}. However, the restriction of 𝑆(𝑡) on 𝑋₂ 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, 𝑛₀∈ℕ^∗, 𝐻>0, and 𝜂>1 such that (i)‖T(𝑛)𝑥‖≥𝐻‖𝑥‖, for all 𝑛∈{0,1,…,𝑛₀} and 𝑥∈𝑌, (ii)‖𝑇(𝑛+𝑛₀)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. Let 𝑥∈𝑌, and 𝑛∈ℕ^∗. Then, there exist 𝑘∈ℕ and 𝑟∈{0,1,…,𝑛₀−1} such that 𝑛=𝑘𝑛₀+𝑟. Therefore, ‖𝑇(𝑛)𝑥‖=‖𝑇(𝑘𝑛₀+𝑟)𝑥‖≥𝜂^𝑘‖𝑇(𝑟)𝑥‖≥𝐻𝜂^𝑘‖𝑥‖. We take 𝜈∶=(1/𝑛₀)ln𝜂>0 to obtain ‖𝑇(𝑛)𝑥‖≥𝐻𝑒^{𝜈𝑘𝑛0}‖𝑥‖=𝐻𝑒^−𝜈𝑟𝑒^{𝜈(𝑘𝑛0+𝑟)}‖𝑥‖>𝐻𝑒^−𝜈𝑛0e^𝜈𝑛‖𝑥‖,∀𝑛∈ℕ.(3.8) For 𝑡≥0, take 𝑛=[𝑡]+1 to have 𝑀𝑒^𝜔‖𝑇(𝑡)𝑥‖≥‖𝑇(𝑛)𝑥‖≥𝐻𝑒^−𝜈𝑛0𝑒^𝜈𝑛‖𝑥‖. Put 𝑁∶=𝐻𝑀⁻¹𝑒^{−(𝜈𝑛0+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, 𝑛₀∈ℕ^∗, 𝐻>0, and 𝜂∈(0,1) such that (i)‖𝑇(𝑛)𝑥‖≤𝐻‖𝑥‖, for all 𝑛∈{0,1,…,𝑛₀} and 𝑥∈𝑌 and (ii)‖𝑇(𝑛+𝑛₀)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 (𝐴,𝑓;x₀): 𝑥̇(𝑡)=𝐴𝑥𝑥((𝑡0))+=𝑓𝑥(₀𝑡),𝑡≥0,.(3.9) If 𝐴 generates the 𝐶₀-semigroup {𝑇(𝑡)}_𝑡≥0, 𝑥∈𝑋 and 𝑓∈𝐿^1loc(ℝ₊,𝑋), then the function 𝑢∶ℝ₊→𝑋 given by 𝑢(𝑡)=𝑇(𝑡)𝑥+^𝑡0𝑇(𝑡−𝑠)𝑓(𝑠)𝑑𝑠(3.10) 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^1loc(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 𝑝_𝑓(𝑛;𝑥)=𝑇(𝑛)𝑥+^𝑛_𝑘=0𝑇(𝑛−𝑘)𝑓(𝑘),(3.11) for each 𝑛∈ℕ.

4. 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 𝑋𝑋_1,ℱ=𝑥∈𝑋∶(𝑇(𝑛)𝑥)_𝑛∈ℕ∈ℱ(𝑋)₁,=𝑥∈𝑋∶lim_𝑡→∞𝑇(𝑡)𝑥=0,(4.1) 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 𝑋=𝑋_1,ℱ⊕𝑋_2,ℱ.(4.2)
We denote by 𝑃₁ the projection onto 𝑋_1,ℱ along 𝑋_2,ℱ and set 𝑃₂=𝐼−𝑃₁ (we will prove that in the case of a no past exponential dichotomy for {𝑇(𝑡)}_𝑡≥0, 𝑋_1,ℱ coincides always with 𝑋₁ and thus the ℱ-independent notation for projectors is consistent).

Remark 4.1. We have that 𝑇(𝑡)𝑋_1,ℱ⊂𝑋_1,ℱ and 𝑇(𝑡)𝑥≠0, for each 𝑥∈𝑋_2,ℱ⧵{0} and 𝑡≥0.

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)|_𝑋2,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 𝑥_𝑓∈𝑋_2,ℱ such that 𝑝_𝑓(⋅;𝑥_𝑓)∈ℱ(𝑋).

Proof. Let 𝑓∈ℰ(𝑋) and 𝑥∈𝑋 from Definition 3.11. For 𝑦=𝑥−𝑃₁𝑥=𝑃₂𝑥 we have that 𝑦∈𝑋_2,ℱ and 𝑝_𝑓(𝑛;𝑦)=𝑝_𝑓(𝑛;𝑥)−𝑇(𝑛)𝑃₁𝑥. Since 𝑝_𝑓(⋅;𝑥)∈ℱ(𝑋) and (𝑇(𝑛)𝑃₁𝑥)∈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 𝑝_𝑓(⋅;𝑧)∈ℱ(𝑋). Since 𝑝_𝑓(𝑛;𝑦)−𝑝_𝑓(𝑛;𝑧)=𝑇(𝑛)(𝑦−𝑧), we have that 𝑦−𝑧∈𝑋_1,ℱ∩𝑋_2,ℱ 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 ‖‖𝑥_𝑓‖‖≤𝐾‖𝑓‖_ℰ(𝑋),‖‖𝑝_𝑓⋅;𝑥_𝑓‖‖_ℱ(𝑋)≤𝐾‖𝑓‖_ℰ(𝑋),(4.3) for all 𝑓∈ℰ(𝑋).

Proof. We define the operator 𝒰∶ℰ(𝑋)⟶𝑋_2,ℱ⊕ℱ(𝑋),𝒰𝑓=𝑥_𝑓,𝑝_𝑓⋅;𝑥_𝑓.(4.4) It is obvious that 𝒰 is a linear operator. Now, we will show that it is also closed.
Let (𝑓_𝑛)_𝑛∈ℕ⊂ℰ(𝑋) such that ‖𝑓_𝑛−𝑓‖_ℰ(𝑋)−−→_𝑛→∞0 and ‖‖𝒰𝑓_𝑛−(𝑦,𝑔)‖‖_{𝑋2,ℱ⊕ℱ(𝑋)}−−−→_𝑛→∞0,(4.5) where 𝑓∈ℰ(𝑋), 𝑦∈𝑋_2,ℱ, and 𝑔∈ℱ(𝑋). For each 𝑛∈ℕ, we take 𝑥_𝑛=𝑥_𝑓𝑛∈𝑋_2,ℱ and 𝑢_𝑛=𝑝_𝑓𝑛(⋅;𝑥_𝑛)∈ℱ(𝑋). We have that ‖𝑥_𝑛−𝑦‖−−→_𝑛→∞0 and ‖𝑢_𝑛−𝑔‖_ℱ(𝑋)−−→_𝑛→∞0.
Since ‖‖𝑢_𝑛(𝑚)−𝑔(𝑚)‖‖≤1𝛽_𝐹(0)‖‖𝑢^𝑛−𝑔‖‖_ℱ(𝑋),(4.6) we have that lim_𝑛→∞𝑢_𝑛(𝑚)=𝑔(𝑘), for all 𝑚∈ℕ. On the other hand, ‖‖𝑢_𝑛(𝑚)−𝑝_𝑓(𝑚;𝑦)‖‖=‖‖‖‖𝑇(𝑚)𝑥_𝑛−𝑦+^𝑚_𝑗=0≤‖𝑇(𝑚)‖‖‖𝑥_𝑛𝑇(𝑚−𝑗)𝑓_𝑛(𝑗)−𝑓(𝑗)‖‖‖‖−𝑦‖‖+^𝑚_𝑗=0‖𝑇(𝑚−𝑗)‖‖‖𝑓_𝑛(𝑗)−𝑓(𝑗)‖‖,(4.7) 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 ‖‖𝑥_𝑓‖‖+‖‖𝑝_𝑓⋅;𝑥_𝑓‖‖_ℱ(𝑋)=‖𝒰𝑓‖_{𝑋2,ℱ⊕ℱ(𝑋)}≤𝐾‖𝑓‖_ℰ(𝑋),(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 (ℰ,ℱ) is admissible to the semigroup {𝑇(𝑡)}_𝑡≥0, then (ℰ,ℓ^∞) is admissible to {𝑇(𝑡)}_𝑡≥0 and ‖𝑝_𝑓(𝑛;𝑥_𝑓)‖≤(𝐾/𝛽_ℱ(0))‖𝑓‖_ℰ(𝑋), for all 𝑓∈ℰ(𝑋) and 𝑛∈ℕ.
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 ℓ¹≠ℰ or ℓ^∞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=𝑋₁.

Proof (Part I). The exponential decay of {𝑇(𝑡)}_𝑡≥0 on 𝑋_1,ℱ.
Let 𝑥∈𝑋_1,ℱ and consider the sequence 𝑓∶ℕ⟶𝑋,𝑓(𝑛)=𝛿₀(𝑛)𝑥,(4.9) which is in ℰ(𝑋) with ‖𝑓‖_ℰ(𝑋)=𝛽_ℰ(0)‖𝑥‖. Observe that 𝑝_𝑓(𝑛;0)=𝑇(𝑛)𝑥, for all 𝑛∈ℕ. Then, 𝑝_𝑓(⋅;0)∈ℱ(𝑋) and from ‖𝑝_𝑓(𝑛;0)‖≤𝐾/(𝛽_𝐹(0))‖𝑓‖_ℰ(𝑋) we obtain ‖𝑇(𝑛)𝑥‖≤𝐾(𝛽_ℰ(0)/𝛽_𝐹(0))‖𝑥‖, for all 𝑛∈ℕ. For 𝑡≥0, taking 𝑛=[𝑡] we have ‖𝑇(𝑡)𝑥‖≤𝑀𝑒^{𝜔(𝑡−𝑛)}‖𝑇(𝑛)𝑥‖≤𝑀𝑒^𝜔𝐾𝛽^ℰ(0)𝛽_𝐹(0)‖𝑥‖.(4.10) Since the constant 𝐶₁∶=𝑀𝑒^𝜔𝐾(𝛽_ℰ(0)/𝛽_𝐹(0)) does not depend on 𝑥, we can write down ‖𝑇(𝑡)𝑥‖≤𝐶₁‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_1,ℱ.(4.11) For 𝑛,𝑚∈ℕ and 𝑥∈𝑋_1,ℱ, we evaluate ‖𝑇(𝑛)𝑥‖ⁿ_𝑗=0𝛿_𝑗(𝑚)=^𝑛_𝑗=0‖𝑇(𝑛−𝑗)𝑇(𝑗)𝑥‖𝛿_𝑗(𝑚)≤𝐶_1𝑛_𝑗=0≤𝐶_1∞‖𝑇(𝑗)𝑥‖𝛿_𝑗(𝑚)_𝑗=0‖𝑇(𝑗)𝑥‖𝛿_𝑗(𝑚)=𝐶₁‖‖𝑝_𝑓(𝑚;0)‖‖,(4.12) which implies ‖𝑇(𝑛)𝑥‖∑^𝑛𝑗=0𝛿_𝑗≤𝐶₁‖𝑝_𝑓(⋅;0)‖ and therefore ‖𝑇(𝑛)𝑥‖‖𝜒_{{0,1,…,𝑛}}‖_ℱ≤𝐶₁‖𝑝_𝑓(⋅;0)‖_ℱ(𝑋).(4.13) By Proposition 4.3 we get that ‖𝑇(𝑛)𝑥‖≤𝐶¹𝐾𝛽_ℰ(0)𝛽_ℱ(𝑛)‖𝑥‖,∀𝑛∈ℕ,𝑥∈𝑋^1,ℱ.(4.14)
If ℓ^∞0⊄ℱ, then 𝛽_ℱ is not bounded and therefore there exists 𝑛₀∈ℕ^∗ such that 𝜂∶=𝐶₁𝐾𝛽_ℰ(0)/𝛽_ℱ(𝑛₀)<1 and ‖𝑇(𝑛₀)𝑥‖≤𝜂‖𝑥‖, for all 𝑥∈𝑋_1,ℱ.
If ℓ^∞0⊂ℱ, then ℰ≠ℓ¹. From Proposition 2.8, it follows that there exists ℎ∈ℰ⧵ℓ¹. Consider 𝛾∶ℕ⟶ℝ,𝛾(𝑛)=^𝑛_𝑗=0||ℎ(𝑗)||,(4.15) which is nondecreasing and lim_𝑛→∞𝛾(𝑛)=∞. For 𝑛∈ℕ and 𝑥∈𝑋_1,ℱ, the sequence 𝑔∶ℕ⟶𝑋,𝑔(𝑚)=𝜒_{{0,1,…,𝑛}}(𝑚)||ℎ(𝑚)||𝑇(𝑚)𝑥(4.16) has finite support, and thus 𝑔∈ℰ(𝑋). Observing that ‖𝑔(𝑚)‖≤𝐶₁‖𝑥‖|ℎ(𝑚)| for every 𝑚∈ℕ, we are led to the evaluation ‖𝑔‖_ℰ(𝑋)≤𝐿‖ℎ‖_ℰ‖𝑥‖. Also, 𝑝_𝑔(𝑚;0)=^𝑚_𝑗=0𝑇(𝑚−𝑗)𝑔(𝑗)=^𝑚_𝑗=0=𝑇(m−𝑗)𝑇(𝑗)𝑥||ℎ(𝑗)||𝜒_{{0,1,…,𝑛}}(𝑗)_𝑚_𝑗=0𝜒_{{0,1,…,𝑛}}(𝑗)||ℎ(𝑗)||𝑇(𝑚)𝑥,(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 𝛾(𝑛)‖𝑇(𝑛)𝑥‖≤𝐾𝐶¹𝛽_ℱ(0)‖ℎ‖^ℰ‖𝑥‖.(4.18) It follows that there exists 𝑛₀∈ℕ^∗ such that 𝜂∶=(𝐾𝐶₁‖ℎ‖_ℰ/𝛽_ℱ(0)𝛾(𝑛₀))<1 and ‖𝑇(𝑛₀)𝑥‖≤𝜂‖𝑥‖, for all 𝑥∈𝑋_1,ℱ.
In both cases, we obtained the existence of some 𝑛₀∈ℕ^∗ and some constant 𝜂∈(0,1) such that ‖𝑇(𝑛₀)𝑥‖≤𝜂‖𝑥‖ for all 𝑥∈𝑋_1,ℱ. Therefore, the semigroup {𝑇(𝑡)_|𝑋1,ℱ}_𝑡≥0 is exponentially stable (see Remark 3.7). Subsequently, there exist 𝑁₁,𝜈₁>0 such that ‖𝑇(𝑡)𝑥‖≤𝑁₁𝑒^−𝜈1𝑡‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_1,ℱ.(4.19)

Proof (Part II). The exponential blow-up of {𝑇(𝑡)}_𝑡≥0 on 𝑋_2,ℱ.
Let 𝑛∈ℕ, 𝑛₀∈ℕ^∗ and 𝑥∈𝑋_2,ℱ⧵{0}, and consider the sequence 𝑓∶ℕ⟶𝑋,𝑓(𝑚)=𝛿_𝑛+𝑛0(𝑚)𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖.(4.20) We have that 𝑓∈ℰ(𝑋) with ‖𝑓‖_ℰ(𝑋)=𝛽_ℰ(0). On the one hand, −^∞_𝑗=_𝑚+1𝛿_𝑛+𝑛0(𝑗)𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖=−^∞_𝑗=0𝛿_𝑛+𝑛0(𝑗)𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖+^𝑚_𝑗=0𝛿_𝑛+𝑛0(𝑘)𝑇(𝑚−𝑗)𝑇(𝑗)𝑥=𝑇(𝑚)−𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖‖‖𝑇𝑛+𝑛₀𝑥‖‖+^𝑚_𝑗=0𝑇(𝑚−𝑗)𝑓(𝑗)=𝑝_𝑓(𝑚;𝑦),(4.21) for all 𝑚∈ℕ (where 𝑦=−𝑥/‖𝑇(𝑛+𝑛₀)𝑥‖∈𝑋_2,ℱ), while on the other hand −^∞_𝑗=_𝑚+1𝛿_𝑛+𝑛0(𝑗)𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖=⎧⎪⎨⎪⎩−𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖,𝑚<𝑛+𝑛⁰0,𝑚≥𝑛+𝑛₀,,(4.22) which implies that 𝑝_𝑓(⋅;𝑦)∈ℱ(𝑋). Then, ‖𝑝_𝑓(𝑚;𝑦)‖≤𝐾/(𝛽_ℱ(0))‖𝑓‖_ℰ(𝑋) and ‖𝑇(𝑚)𝑥‖≤𝐾𝛽^ℰ(0)𝛽_ℱ(0)‖‖𝑇𝑛+𝑛⁰𝑥‖‖,∀0≤𝑚<𝑛+𝑛₀.(4.23) Taking now 𝑚=𝑛=0 we can write down ‖‖𝑇𝑛₀𝑥‖‖≥𝐾⁻¹𝛽_ℱ(0)𝛽_ℰ(0)‖𝑥‖,∀𝑛⁰∈ℕ^∗,𝑥∈𝑋_2,ℱ.(4.24) For 𝑡≥0, we put 𝑛₀=[𝑡]+1 to evaluate 𝑀𝑒^𝜔‖𝑇(𝑡)𝑥‖≥‖𝑇(𝑛₀)𝑥‖≥𝐾⁻¹(𝛽_ℱ(0)/𝛽_ℰ(0))‖𝑥‖. Denoting 𝐶₂∶=(𝐾𝑀𝑒^𝜔)⁻¹𝛽_ℱ(0)/𝛽_ℰ(0) we have that ‖𝑇(𝑡)𝑥‖≥𝐶₂‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_2,ℱ.(4.25) Also, from (4.23), taking 𝐶^∶=1+𝐾(𝛽_ℰ(0)/𝛽_ℱ(0)) we have that 𝐶^{‖𝑇(ℎ)𝑥‖}≥‖𝑇(𝑛)𝑥‖,∀ℎ≥𝑛,𝑥∈𝑋_2,ℱ.(4.26) Since 𝑛,𝑛₀, and 𝑥 are randomly taken (see the beginning of Part II), we have that ‖𝑇(𝑛)𝑥‖‖‖𝑇𝑛+𝑛₀𝑥‖‖^{𝑛+𝑛0−1}_𝑗=𝑛𝛿_𝑗(𝑚)≤𝐶′^{𝑛+𝑛0−1}_𝑗=𝑛‖𝑇(𝑗)𝑥‖‖‖𝑇𝑛+𝑛₀𝑥‖‖𝛿^𝑗=𝐶′_{𝑛+𝑛0−1}(𝑚)_𝑗=𝑛≤𝐶′‖‖𝑝_𝑓‖‖𝑝_𝑓(𝑗;𝑦)‖‖𝛿_𝑗(𝑚)(𝑚;𝑦)‖‖,∀𝑚∈ℕ,(4.27) and therefore ‖𝑇(𝑛)𝑥‖/‖𝑇(𝑛+𝑛₀)𝑥‖𝜒_{{𝑛,𝑛+1,…,𝑛+𝑛0−1}}≤𝐶′‖𝑝_𝑓(⋅;𝑦)‖. It follows that ‖𝑇(𝑛)𝑥‖‖‖𝑇𝑛+𝑛₀𝑥‖‖‖‖𝑅^𝑛𝜒_{{0,1,…,𝑛0−1}}‖‖_ℱ≤𝐶′‖‖‖‖𝑝_𝑓(⋅;𝑦)‖‖‖‖_ℱ,(4.28) or equivalently, ‖𝑇(𝑛)𝑥‖‖‖𝑇𝑛+𝑛₀𝑥‖‖𝛽^ℱ𝑛₀−1≤𝐶′‖‖𝑝_𝑓(⋅;𝑦)‖‖_ℱ(𝑋).(4.29) Using the fact that (4.29)‖𝑝_𝑓(⋅;𝑦)‖_ℱ(𝑋)≤𝐾‖𝑓‖_ℰ(𝑋), we deduce that ‖‖𝑇𝑛+𝑛₀𝑥‖‖≥𝛽^ℱ𝑛₀−1𝐶′𝐾𝛽_ℰ(0)‖𝑇(𝑛)𝑥‖,∀𝑛∈ℕ,𝑛⁰∈ℕ^∗,𝑥∈𝑋_2,ℱ.(4.30)
If ℓ^∞0⊄ℱ, then 𝛽_ℱ is not bounded and therefore there exists 𝑛₀∈ℕ^∗ such that 𝜂∶=(𝛽_ℱ(𝑛₀−1))/(𝐶^𝐾𝛽_ℰ(0))>1 and ‖𝑇(𝑛+𝑛₀)𝑥‖≥𝜂‖𝑇(𝑛)𝑥‖ for all 𝑛∈ℕ and 𝑥∈𝑋_2,ℱ.
If ℓ^∞0⊂ℱ, then ℰ≠ℓ¹ and therefore there exists ℎ∈ℰ⧵ℓ¹. Consider 𝛾 as in (4.15), and for 𝑛,𝑛₀∈ℕ and 𝑥∈𝑋_2,ℱ⧵{0} we define 𝑔∶ℕ⟶𝑋,𝑔(𝑚)=^𝑛0_𝑗=0𝛿_𝑛+𝑗(𝑚)||ℎ(𝑗)||𝑇(𝑛+𝑗)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖.(4.31) Since ‖𝑔(𝑚)‖≤𝐶′𝜒_{{𝑛,𝑛+1,…,𝑛+𝑛0}}(𝑚)|(𝑅^𝑛ℎ)(𝑚)|≤𝐶′|(𝑅^𝑛ℎ)(𝑚)| for each 𝑚∈ℕ, we have that 𝑔∈ℰ(𝑋) with ‖𝑔‖_ℰ(𝑋)≤𝐶′‖𝑅^𝑛ℎ‖_ℰ=𝐶′‖ℎ‖_ℰ. Note that 𝑔(𝑚)=𝜒_{{𝑛,𝑛+1,…,𝑛+𝑛0}}(𝑚)|𝑅^𝑛ℎ(𝑚)|𝑇(𝑚)𝑥/‖𝑇(𝑛+𝑛₀)𝑥‖, for all 𝑚∈ℕ. On the one hand, −^∞_𝑗=_𝑚+1𝜒_{{𝑛,…,𝑛+𝑛0}}(𝑗)||𝑅^𝑛ℎ(𝑗)||𝑇(𝑚)𝑥=−_∞‖‖𝑇𝑛+𝑛₀𝑥‖‖_𝑗=0𝜒_{{𝑛,…,𝑛+𝑛0}}(𝑗)||𝑅^𝑛ℎ(𝑗)||𝑇(𝑚)𝑥+_𝑚‖‖𝑇𝑛+𝑛₀𝑥‖‖+_𝑗=0𝑇(𝑚−𝑗)𝜒_{{𝑛,…,𝑛+𝑛0}}(𝑗)||𝑅^𝑛ℎ(𝑗)||𝑇(𝑗)𝑥=𝑇(𝑚)𝛾𝑛₀‖‖𝑇𝑛+𝑛₀𝑥‖‖−𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖+^𝑚_𝑗=0=𝑝_𝑔𝑇(𝑚−𝑗)𝑔(𝑗)(𝑚;𝑧),(4.32) for all 𝑚∈ℕ (where 𝑧∶=𝛾(𝑛₀)(−𝑥/‖𝑇(𝑛+𝑛₀)𝑥‖)∈𝑋_2,ℱ), while on the other hand −^∞_𝑗=_𝑚+1𝜒_{{𝑛,…,𝑛+𝑛0}}(𝑗)||𝑅^𝑛ℎ(𝑗)||𝑇(𝑚)𝑥=⎧⎪⎪⎨⎪⎪⎩−𝛾𝑛₀‖‖𝑇𝑛+𝑛₀𝑥‖‖𝑇(𝑚)𝑥−𝛾𝑛‖‖𝑇𝑛+𝑛₀𝑥‖‖,𝑚<𝑛,₀−𝛾(𝑚−𝑛)𝑇(𝑚)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖‖,𝑛≤𝑚<𝑛+𝑛⁰0,𝑚≥𝑛+𝑛₀,.(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 𝛾𝑛₀−||ℎ(0)||‖𝑇(𝑛)𝑥‖‖𝑇𝑛+𝑛₀𝑥‖≤𝛽𝐾^ℱ𝐶(0′)‖ℎ‖^ℰ.(4.34) Thus, ‖𝑇(𝑛+𝑛₀)𝑥‖≥(𝛽_ℱ(0)/𝐾𝐶^‖ℎ‖_ℰ)(𝛾(𝑛₀)−|ℎ(0)|)‖𝑇(𝑛)𝑥‖, for all 𝑛,𝑛₀∈ℕ and 𝑥∈𝑋_2,ℱ. Then, there exist 𝑛₀∈ℕ^∗ and 𝜂>1 such that ‖‖𝑇𝑛+𝑛₀𝑥‖‖≥𝜂‖𝑇(𝑛)𝑥‖,∀𝑛∈ℕ,𝑥∈𝑋_2,ℱ.(4.35)
In both cases, we obtained the existence of some 𝑛₀∈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 𝑁₂,𝜈₂>0 such that ‖𝑇(𝑡)𝑥‖≥𝑁₂𝑒^𝜈2𝑡‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_2,ℱ.(4.36)
If we take 𝜈∶=min{𝜈₁,𝜈₂}>0, all the conditions guaranteeing the existence of a no past exponential dichotomy are met.

Proof (Part III). We prove that 𝑋_1,ℱ=𝑋₁, no matter how we choose the sequence Schäffer space ℱ. If 𝑥∈𝑋_1,ℱ, then ‖𝑇(𝑡)𝑥‖≤𝑁₁𝑒^−𝜈1𝑡‖𝑥‖, for all 𝑡≥0, which implies 𝑥∈𝑋₁.
Conversely, let 𝑥∈𝑋₁, 𝑢∈𝑋_1,ℱ, and 𝑣∈𝑋_2,ℱ such that 𝑥=𝑢+𝑣. For every 𝑛∈ℕ, we have that ‖𝑇(𝑛)𝑥‖=≥‖𝑁𝑇₂(𝑛)𝑣−(−𝑇(𝑛)𝑢)‖≥‖𝑇(𝑛)𝑣‖−‖𝑇(𝑛)𝑢‖𝑒^𝜈2𝑛‖𝑣‖−𝑁₁𝑒^−𝜈1𝑛‖𝑢‖.(4.37) If we suppose that 𝑣≠0, then ‖𝑇(𝑛)𝑥‖−−→_𝑛→∞∞ contradicting the fact that 𝑥∈𝑋₁. 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 (ℓ^𝑝,ℓ^𝑞)1≤𝑝≤𝑞≤∞ and (𝑝,𝑞)≠(1,∞) and the hyperbolicity of a 𝐶₀-semigroup {𝑇(𝑡)}_𝑡≥0, when we assume a priori that the kernel of the splitting projection is 𝑇(𝑡)-invariant and 𝑇(𝑡)|_Ker𝑃 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)|_Ker𝑃 is invertible (we do not even assume that Ker𝑃 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 discrete-time mild solutions” (i.e., the solution of the inhomogeneous difference equation problem) to stay in ℓ^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 ℓ^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 ℓ¹≠ℰ or ℓ^∞0⊄ℱ, then the subspace 𝑋_1,ℱ (which induces the no past exponential dichotomy) is actually the regular stable subspace 𝑋₁. If we would impose in addition that the complement of 𝑋₁ (denoted by 𝑋₂) is also 𝑇(𝑡)-invariant, then the aforementioned admissibility condition would imply that {𝑇(𝑡)}_𝑡≥0 extends automatically to a 𝐶₀-group on 𝑋₂, 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 𝑋₂” converts to “hyperbolicity”.

The next example shows that the condition “𝐸≠ℓ¹ or ℓ^∞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∈ℓ¹, then there exists 𝑥=−∑^∞𝑘=0𝑓(𝑘)∈ℝ (the series being absolutely convergent) such that 𝑝_𝑓(⋅;𝑥)∈ℓ^∞0(𝑋). Therefore, the pair (ℓ¹,ℓ^∞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.
The above theorem shows that the space 𝑋_1,ℱ induces a no past exponential dichotomy for {𝑇(𝑡)}_𝑡≥0—an exponentially bounded semigroup—and that 𝑋_1,ℱ is actually 𝑋₁. Concerning the hypothesis over ℰ and ℱ 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: (𝐵₁,𝐷₁) is said to be stronger than (𝐵,𝐷) (or the later pair is weaker than the former) if 𝐵↪𝐵₁ and 𝐷₁↪𝐷. In the case of sequence Schäffer spaces the two inclusions with continuous injection are equivalent with the corresponding algebraic inclusion (Proposition 2.9); so the pair (ℓ¹,ℓ^∞0) is stronger than (ℰ,ℱ) if ℰ⊂ℓ¹ and ℓ^∞0⊂ℱ, or equivalently (considering Proposition 2.8), ℰ=ℓ¹ and ℓ^∞0⊂ℱ. Therefore, the hypothesis “𝐸≠ℓ¹ or ℓ^∞0⊄ℱ” can be replaced by, “the pair (ℓ¹,ℓ^∞0) is not stronger than the pair (ℰ,ℱ)”.
Dropping off the restriction “ℓ¹≠ℰ or ℓ^∞0⊄ℱ”, the proof of Theorem 4.5 still provides useful information. More accurately, the admissibility of any pair of sequence Schäffer spaces implies the existence of an ordinary dichotomy for the semigroup {𝑇(𝑡)|_𝑋1,ℱ}_𝑡≥0 (in the sense of Definition 3.6).

Corollary 4.9. Let (ℰ,ℱ) be a pair sequence Schäffer space and, {𝑇(𝑡)}_𝑡≥0 be an exponentially bounded semigroup. If the pair (ℰ,ℱ) is admissible to {𝑇(𝑡)}_𝑡≥0, then {𝑇(𝑡)}_𝑡≥0 has an ordinary dichotomy; that is, there exist two constants 𝐶₁,𝐶₂>0 such that ‖‖𝑇𝑇((𝑡𝑡))𝑥𝑥‖‖≤≥𝐶𝐶₁₂‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_1,ℱ,‖𝑥‖,∀𝑡≥0,𝑥∈𝑋_2,ℱ.(4.38)

Proof. Note that in the proof of Theorem 4.5, we do not use the hypothesis “𝐸≠ℓ¹ or ℓ^∞0⊄ℱ” to prove the relations (4.11) and (4.25).

Corollary 4.10. Let 𝑝,𝑞∈[1,∞] such that (𝑝,𝑞)≠(1,∞) and let {𝑇(𝑡)}_𝑡≥0 be an exponentially bounded semigroup. If the pair (ℓ^𝑝,ℓ^𝑞) is admissible to {𝑇(𝑡)}_𝑡≥0, then {𝑇(𝑡)}_𝑡≥0 has a no past exponential dichotomy.

Proof. We have the equivalence (due to Proposition 2.10 and the fact that 𝛽_ℓ^𝑞(𝑛)=(𝑛+1)^1/𝑞 for each 𝑛∈ℕ): ℓ^∞0⊂ℓ^𝑞 if and only if 𝑞=∞. Thus, given (𝑝,𝑞)≠(1,∞) the hypothesis of Theorem 4.5 is satisfied and the conclusion is immediate.

Corollary 4.11. Let ℓ^Φ be a sequence Orlicz space and let {𝑇(𝑡)}_𝑡≥0 be an exponentially bounded semigroup. If the pair (ℓ^Φ,ℓ^Φ) is admissible to {𝑇(𝑡)}_𝑡≥0, then {𝑇(𝑡)}_𝑡≥0 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 𝑇(𝑡)-invariance of 𝑋₂, we can deduce the invertibility of 𝑇(𝑡)|_𝑋2, thus obtaining the exponential dichotomy (or equivalently, hyperbolicity) for {𝑇(𝑡)}_𝑡≥0. Note that the first condition of the following theorem does not require that 𝑋₂ is invariant to all operators 𝑇(𝑡). However, we can prove that {𝑇(𝑡)|_𝑋2}_𝑡≥0 is a semigroup (exponentially bounded if {𝑇(𝑡)}_𝑡≥0 is exponentially bounded); hence the invertibility of all its operators follows from the invertibility of just one of them.

Theorem 4.12. Let ℰ and ℱ be two sequence Schäffer spaces such that ℓ¹≠ℰ or ℓ^∞0⊄ℱ and let {𝑇(𝑡)}_𝑡≥0 be an exponentially bounded semigroup. If(i)there exists 𝑡₀>0 such that 𝑃₂𝑇(𝑡)𝑃₂=𝑇(𝑡)𝑃₂ for all 𝑡∈[0,𝑡₀], (ii)(ℰ,ℱ) is admissible to 𝑇, then 𝑇(𝑡)∶𝑋₂→𝑋₂ is invertible for each 𝑡≥0. Therefore, {𝑇(𝑡)}_𝑡≥0 has an exponential dichotomy.

Proof. Let 𝑡>𝑡₀ and consider 𝑛∈ℕ and 𝛿∈[0,𝑡₀) such that 𝑡=𝑛𝑡₀+𝛿. If 𝑥∈𝑋₂, we have that 𝑇(𝑡)𝑥=𝑇(𝛿)𝑇(𝑡₀)^𝑛𝑥∈𝑋₂. Therefore, the condition (i) assures that the operator 𝑇(𝑡)∶𝑋₂→𝑋₂ 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, 𝑛₀=[𝑡]+1, 𝑦∈𝑋₂ and consider the sequence 𝑓∶ℕ⟶𝑋,𝑓(𝑛)=−𝛿_𝑛0(𝑛)𝑇𝑛₀−𝑡𝑦,(4.39) which belongs to ℰ(𝑋) with ‖𝑓‖_ℰ(𝑋)=𝛽_ℰ(0)‖𝑇(𝑛₀−𝑡)𝑦‖. Then, according to Proposition 4.2, there exists a unique 𝑥∈𝑋₂ such that 𝑝_𝑓(⋅;𝑥)∈ℱ(𝑋). Since 𝑝_𝑓(𝑛;𝑥)=𝑇(𝑛)𝑥−^𝑛_𝑘=0=𝑇(𝑛)𝑥−𝑇𝛿𝑛−_𝑛0(𝑘𝑛₀)𝑇(𝑛−𝑘)𝑇𝑛₀−𝑡𝑦=𝑇(𝑛−𝑡)(𝑇(𝑡)𝑥−𝑦)𝑇,𝑛₀−𝑡𝑦=𝑇(𝑛−𝑡)𝑇(𝑡)𝑥−𝑇(𝑛−𝑡)𝑦(4.40) for all 𝑛∈ℕ, 𝑛≥𝑛₀, we have that 𝑇(𝑡)𝑥,𝑦∈𝑋₂ and ‖‖𝑝_𝑓(𝑛;𝑥)‖‖=‖𝑇(𝑛−𝑡)(𝑇(𝑡)𝑥−𝑦)‖≥𝑁₂𝑒^{𝜈2(𝑛−𝑡)}‖𝑇(𝑡)𝑥−𝑦‖,∀𝑛≥𝑛₀.(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)∈ℓ^∞(𝑋). It follows that 𝑇(𝑡)𝑥=𝑦. We proved that 𝑇(𝑡)|_𝑋2 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 𝑋₂ 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 {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 ‖𝐿𝑓‖_ℱ=‖𝑅𝐿𝑓‖_ℱ≤‖𝑓‖_ℱ.(4.42)

Theorem 4.14. 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.

Proof. We have that there exist 𝑁,𝜈>0 such that ‖‖𝑇(𝑡)𝑃₁𝑥‖‖≤𝑁𝑒^−𝜈𝑡‖‖𝑃₁𝑥‖‖,‖‖𝑇(𝑡)^−1|Im𝑃₂𝑃₂𝑥‖‖≤𝑁𝑒^−𝜈𝑡‖‖𝑃₂𝑥‖‖,(4.43) for all 𝑡≥0 and 𝑥∈𝑋.
Let 𝑓∈ℰ(𝑋) and consider the sequence 𝜑∶ℕ⟶𝑋,𝜑(𝑛)=^𝑛_𝑘=0𝑇(𝑛−𝑘)𝑃₁𝑓(𝑘)−^∞_𝑘_=𝑛+1𝑇(𝑘−𝑛)⁻¹𝑃₂𝑓(𝑘),(4.44) which is well defined, since 𝑓∈ℓ^∞(𝑋) and ^∞_𝑘=𝑛+1‖‖𝑇(𝑘−𝑛)⁻¹𝑃₂𝑓(𝑘)‖‖≤𝑁‖‖𝑃₂‖‖^∞_𝑘_=𝑛+1≤𝑁‖‖𝑃𝑒^{−𝜈(𝑘−𝑛)}‖𝑓(𝑘)‖₂‖‖‖𝑓‖_∞1−𝑒^−𝜈,∀𝑛∈ℕ.(4.45) For any 𝑛∈ℕ, we have the following evaluation: ‖𝜑(𝑛)‖≤𝑁‖‖𝑃₁‖‖^𝑛_𝑘=0𝑒^{−𝜈(𝑛−𝑘)}‖𝑓(𝑘)‖+𝑁‖‖𝑃₂‖‖^∞_𝑘_=𝑛+1≤𝑁‖‖𝑃₁𝑒^{−𝜈(𝑘−𝑛)}‖𝑓(𝑘)‖,‖‖+‖‖𝑃₂‖‖^𝑛_𝑘=0𝑒^−𝜈𝑘‖‖𝑅^𝑘𝑓‖‖(𝑛)+^∞_𝑘=1𝑒^−𝜈𝑘‖‖𝐿^𝑘𝑓‖‖(𝑛).(4.46) From the hypothesis and Remark 4.13 we have that 𝑓,𝑅^𝑘𝑓,𝐿^𝑘𝑓∈ℱ(𝑋) with ‖𝐿^𝑘𝑓‖_ℱ(𝑋)≤‖𝑓‖_ℱ(𝑋) for all 𝑘∈ℕ. Therefore, ^∞_𝑘=1‖‖𝑒^−𝜈𝑘‖‖𝐿^𝑘𝑓‖‖‖‖_ℱ≤^∞_𝑘=1𝑒^−𝜈𝑘‖𝑓‖_ℱ(𝑋)=𝑒^−𝜈1−𝑒^−𝜈‖𝑓‖_ℱ(𝑋),_∞_𝑘=0‖‖𝑒^−𝜈𝑘‖‖𝑅^𝑘𝑓‖‖‖‖_ℱ=^∞_𝑘=0𝑒^−𝜈𝑘‖𝑓‖_ℱ(𝑋)=11−𝑒^−𝜈‖𝑓‖_ℱ(𝑋),(4.47) and thus 𝑔∶=∑^𝑛𝑘=0𝑒^−𝜈𝑘‖𝑅^𝑘𝑓‖+∑^∞𝑘=1𝑒^−𝜈𝑘‖𝐿^𝑘𝑓‖ exists as an element in ℱ.
From (4.46), denoting 𝐶∶=𝑁(‖𝑃₁‖+‖𝑃₂‖) 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 𝑇(𝑘−𝑛)⁻¹𝑥=𝑇(𝑛)𝑇(𝑘)⁻¹𝑥 for all 𝑥∈𝑋₂.
Then, 𝜑(𝑛)=^𝑛_𝑘=0𝑇(𝑛−𝑘)𝑃₁𝑓(𝑘)−^∞_𝑘=0+𝑇(𝑛)𝑇(𝑘)⁻¹𝑃₂𝑓(𝑘)_𝑛_𝑘=0=𝑇(𝑛)𝑇(−𝑛−_∞𝑘)𝑇(𝑘)𝑇(𝑘)⁻¹𝑃₂𝑓(𝑘)_𝑘=0𝑇(𝑘)⁻¹𝑃₂𝑓(𝑘)+^𝑛_𝑘=0=𝑇(𝑛)𝑥+_𝑛𝑇(𝑛−𝑘)𝑃₁𝑓(𝑘)+𝑃₂𝑓(𝑘)_𝑘=0𝑇(𝑛−𝑘)𝑓(𝑘)=𝑝_𝑓(𝑛;𝑥),∀𝑛∈ℕ,(4.48) where 𝑥=−∑^∞𝑘=0𝑇(𝑘)⁻¹𝑃₂𝑓(𝑘)∈𝑋₂ (the 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). 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 𝑡₀>0 such that 𝑃₂𝑇(𝑡)𝑃₂=𝑇(𝑡)𝑃₂ for all 𝑡∈[0,𝑡₀];(ii)ℰ⊂ℱ;(iii)𝐿ℰ⊂ℰ or 𝐿ℱ⊂ℱ;(iv)ℓ¹≠ℰ or ℓ^∞0⊄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^−𝑡𝑥. Clearly, {𝑇(𝑡)}_𝑡≥0 defines a 𝐶₀-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→ℝ𝑓(𝑛)=1/(𝑛+1). We have that 𝑓∈ℓ²⧵ℓ¹, and for any 𝑥∈ℝ, 𝑝_𝑓(𝑛;𝑥)=𝑇(𝑛)𝑥+^𝑛_𝑘=0𝑇(𝑛−𝑘)𝑓(𝑘)=𝑒^−𝑛𝑥+^𝑛_𝑘=0𝑒^{−(𝑛−𝑘)}1𝑘+1.(4.49) It follows that |𝑝_𝑓(𝑛;𝑥)|≥1/(𝑛+1)−𝑒^−𝑛|𝑥|, for all 𝑛∈ℕ. Then, ^∞_𝑛=0||𝑝_𝑓(𝑛;𝑥)||≥^∞_𝑛=01𝑛+1−1−|𝑥𝑒|⁻¹.(4.50) We obtained that 𝑝_𝑓(⋅;𝑥)∉ℓ¹ for any 𝑥∈𝑋. Thus, the pair (ℓ²,ℓ¹) 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 𝑋₂={0}, and therefore, the invariance condition of either ℰ or ℱ under the left shift is not necessary.

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 𝑃₁=𝐼. With this remark, observe that the expression of 𝜑 in (4.44) reduces to 𝜑(𝑛)=^𝑛_𝑘=0𝑇(𝑛−𝑘)𝑓(𝑘),(4.51) and, respectively, the evaluation from (4.46) becomes ‖𝜑(𝑛)‖≤𝑁^𝑛_𝑘=0𝑒^−𝜈𝑘‖‖𝑅^𝑘𝑓‖‖(𝑛)≤𝑁^∞_𝑘=0𝑒^−𝜈𝑘‖‖𝑅^𝑘𝑓‖‖(𝑛).(4.52) The sequence 𝑔∶=∑^∞𝑘=0𝑒^−𝜈𝑘‖𝑅^𝑘𝑓‖ 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 (ℓ^𝑝,ℓ^𝑞) is admissible to {𝑇(𝑡)}_𝑡≥0.

Proof. The spaces ℓ^𝑝 (1≤𝑝≤∞) are invariant under the left shift, and with 𝑝≤𝑞 we also have that ℓ^𝑝⊂ℓ^𝑞. Applying now Theorem 4.14 we obtain the previously statement.

Corollary 4.20. Let ℓ^Φ be a sequence Orlicz space. The semigroup {𝑇(𝑡)}_𝑡≥0 is exponentially dichotomic if and only if the pair (ℓ^Φ,ℓ^Φ) is admissible to {𝑇(𝑡)}_𝑡≥0.

Proof. It follows immediately from Theorem 4.14 and Corollary 4.11, since any ℓ^Φ is invariant under the left shift.