# Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

Authors: Răzvan O. Moşincat; Ciprian Preda; Petre Preda
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

We analyze the existence of (no past) exponential dichotomies
for a well-posed autonomous differential equation (that generates aC
0-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.


---

## Body

## 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
)
=
A
x
(
t
) (assuming its well-posedness, i.e., A generates a C
0-semigroup {
T
(
t
)
}
t
≥
0) consists in the existence of a bounded projection P such that the solutions that start in X
1
=
Im
P decay to zero and the solutions that start in X
2
=
Im
(
I
-
P
) are unbounded. In the hypothesis that X
2 is T
(
t
)-invariant and finite-dimensional, the existence of a dichotomy for (
A
) implies that solutions starting in X
2 exist in backward time (or equivalently, {
T
(
t
)
} extends to a C
0-group on X
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 T
(
t
)
|
X
2 are invertible (we do not even assume that X
2 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

Letℕ 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
(2.1)
R
f
(
n
)
=
{
f
(
n
-
1
)
,
n
∈
N
*
,
0
,
n
=
0
,
L
f
(
n
)
=
f
(
n
+
1
)
,
known as the right shift operator, respectively, the left shift operator. A simple verification gives us L
R
f
=
f and R
L
f
(
n
)
=
f
(
n
) for n
∈
ℕ
*, R
L
f
(
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:(
s
1
)
δ
0
∈
ℰ,
(
s
2
)
iff
∈
ℰ, then R
f
∈
ℰ and ∥
R
f
∥
ℰ
=
∥
f
∥
ℰ,
(
s
3
)
iff
∈
𝒮 and g
∈
ℰ such that |
f
|
≤
|
g
|, then f
∈
ℰ and ∥
f
∥
ℰ
≤
∥
g
∥
ℰ.Remark 2.2.
By(
s
1
) and (
s
2
) we have that any sequence with finite support is contained in any sequence Schäffer space, hence χ
{
0,1
,
…
,
n
}
∈
ℰ for any sequence Schäffer space ℰ and n
∈
ℕ. The third property is called the ideal property and will play a central role in our investigations.Example 2.3.
Common instances of sequence Schäffer spaces are the spaces ofp-summable sequences, namely, for p
∈
[
1
,
∞
),
(2.2)
l
p
=
{
f
:
N
→
R
:
∑
k
=
0
∞
|
f
(
k
)
|
p
<
∞
}
,
with
the
norm
⁡
‖
f
‖
p
=
(
∑
k
=
0
∞
|
f
(
k
)
|
p
)
1
/
p
,
l
∞
=
{
f
:
N
→
R
:
sup
⁡
n
∈
N
|
f
(
n
)
|
<
∞
}
,
with
the
norm
⁡
‖
f
‖
∞
=
sup
⁡
n
∈
N
|
f
(
n
)
|
.
The subspace of ℓ
∞, ℓ
0
∞
=
{
f
∈
ℓ
∞
:
lim
n
→
∞
f
(
n
)
=
0
} (often denoted by c
0) with the induced norm is another example of sequence Schäffer space.
It is easy to check that(
c
,
∥
·
∥
∞
) (the space of all convergent sequences) is not a sequence Schäffer space.
The spacesℓ
1, 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
(2.3)
α
E
(
n
)
=
inf
⁡
{
L
>
0
:
∑
k
=
0
n
|
f
(
k
)
|
≤
L
‖
f
‖
E
,
∀
f
∈
E
}
,
β
E
(
n
)
=
‖
χ
{
0,1
,
…
,
n
}
‖
E
,
which are both nondecreasing and β
ℰ
(
n
)
>
0, for all n
∈
ℕ.Example 2.4.
Other remarkable examples of sequence Schäffer spaces are thesequence 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
)
=
∫
0
t
φ
(
s
)
d
s
,
t
≥
0. Consider
(2.4)
l
Φ
=
{
f
∈
S
:
exists
c
>
0
such
that
∑
k
=
0
n
Φ
(
c
|
f
(
k
)
|
)
<
∞
}
with
the
norm
⁡
‖
f
‖
Φ
=
inf
⁡
{
c
>
0
∑
k
=
0
n
Φ
(
c
-
1
|
f
(
k
)
|
)
≤
1
}
(
the
Luxemburg
norm
⁡
)
.
For the Banach space (
ℓ
Φ
,
∥
·
∥
Φ
) the conditions (
s
1
), (
s
2
), and (
s
3
) are verified; hence (
ℓ
Φ
,
∥
·
∥
Φ
) is a sequence Schäffer space.
For1
≤
p
<
∞, taking φ
(
t
)
=
p
t
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.
Letp
∈
[
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
(2.5)
Φ
-
1
(
1
n
+
1
)
=
(
1
n
+
1
)
1
/
p
,
∀
n
∈
N
.
Let x
∈
(
0,1
] and m
=
[
1
/
x
]
∈
ℕ
*. Using the fact that Φ
-
1 is nondecreasing, we have that
(2.6)
(
1
m
+
1
)
1
/
p
=
Φ
-
1
(
1
m
+
1
)
≤
Φ
-
1
(
x
)
≤
Φ
-
1
(
1
m
)
=
(
1
m
)
1
/
p
,
which implies that
(2.7)
[
1
(
[
1
/
x
]
+
1
)
x
]
1
/
p
≤
Φ
-
1
(
x
)
x
1
/
p
≤
[
1
x
[
1
/
x
]
]
1
/
p
,
for all x
∈
(
0,1
]. Hence lim
x
→
0
Φ
-
1
(
x
)
x
-
(
1
/
p
)
=
1 and
(2.8)
lim
⁡
u
→
0
Φ
(
u
)
u
p
=
lim
⁡
u
→
0
1
[
Φ
-
1
(
Φ
(
u
)
)
/
(
Φ
(
u
)
)
1
/
p
]
p
=
1
.
.Example 2.6.
Considerφ
:
ℝ
+
→
ℝ
+:
(2.9)
φ
(
t
)
=
∑
m
=
1
∞
t
m
m
2
,
Φ
(
t
)
=
∫
0
t
φ
(
s
)
d
s
=
∑
m
=
1
∞
t
1
+
1
/
m
m
(
m
+
1
)
.
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(2.10)
α
l
p
(
n
)
=
(
n
+
1
)
1
-
1
/
p
,
β
l
p
(
n
)
=
(
n
+
1
)
1
/
p
for 1
≤
p
≤
∞ (with the convention 1
/
∞
=
0) and for the Orlicz sequence spaces,
(2.11)
α
l
Φ
(
n
)
=
(
n
+
1
)
Φ
-
1
(
1
n
+
1
)
,
β
l
Φ
(
n
)
=
1
Φ
-
1
(
1
/
(
n
+
1
)
)
.
For two Banach spaces(
B
1
,
∥
·
∥
1
) and (
B
2
,
∥
·
∥
2
), we say that B
1 is continuously embedded in B
2 (B
1
↪
B
2) if B
1
⊂
B
2 and there exists c
>
0 such that ∥
f
∥
2
≤
c
∥
f
∥
1 for all f
∈
B
1. 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 ℓ
1
↪
ℰ
↪
ℓ
∞ with β
ℰ
(
0
)
∥
f
∥
∞
≤
∥
f
∥
ℰ for all f
∈
ℰ and ∥
f
∥
ℰ
≤
β
ℰ
(
0
)
∥
f
∥
1 for all f
∈
ℓ
1.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
=
ℓ
1 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)
iff
∈
𝒮
(
X
) has finite support, then f
∈
ℰ
(
X
);
(ii)
iff
∈
ℰ
(
X
), then R
f
∈
ℰ
(
X
) and ∥
R
f
∥
E
(
X
)
=
∥
f
∥
ℰ
(
X
);
(iii)
iff
∈
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 notationℬ
(
X
) for the class of bounded linear operators acting on X.


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

LetX be a Banach space, x
0
∈
X, A
:
D
(
A
)
⊂
X
→
X a linear operator, and consider the (abstract) Cauchy problem (
A
;
x
0
): (3.1)
x
̇
(
t
)
=
A
x
(
t
)
,
t
≥
0
,
x
(
0
)
=
x
0
.
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
;
x
0
). 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 C
0-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 anyt
,
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
)
∥
≤
M
e
ω
t for all t
≥
0,
then{
T
(
t
)
}
t
≥
0 is said to be exponentially bounded.IfY is a closed subspace of X and {
T
(
t
)
}
t
≥
0 is a (C
0-)semigroup such that Y is T
(
t
)-invariant for any t
≥
0, then the restrictions T
(
t
)
|
Y form a (C
0-)semigroup called the subspace (C
0-)semigroup on Y ([18, page 43]). A (C
0-)semigroup can be extended to a (C
0-)group if and only if there exists t
0
>
0 such that T
(
t
0
) is invertible (see [18, page 80] or [19, Section  1.6]).The linear operator defined by(3.2)
D
(
A
)
=
{
x
∈
X
:
lim
⁡
t
↓
0
T
(
t
)
x
-
x
t
exists
}
,
A
x
=
lim
⁡
t
↓
0
T
(
t
)
x
-
x
t
for
x
∈
D
(
A
)
is the infinitesimal generator of the semigroup {
T
(
t
)
}
t
≥
0. If {
T
(
t
)
}
t
≥
0 is a C
0-semigroup, then A is closed and densely defined and {
T
(
t
)
}
t
≥
0 is exponentially bounded (for details, see [19, Section  1.2]). It is clear that a classical solution for (
A
;
x
) with A being the infinitesimal generator of {
T
(
t
)
}
t
≥
0 exists only if x
∈
D
(
A
) in which case T
(
·
)
x is the unique classical solution; otherwise it is said to be the mild solution of (
A
;
x
).A linear and bounded operatorT acting on a complex Banach space is said to be hyperbolic if σ
(
T
)
∩
Γ
=
∅ (where Γ
=
{
z
∈
ℂ
:
|
z
|
=
1
} denotes the unit circle in the complex plane and σ
(
T
) is the spectrum of T). The spectral Riesz projection P for a hyperbolic operator T is given by(3.3)
P
=
1
2
π
i
∫
Γ
(
λ
I
-
T
)
-
1
d
λ
.
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 (3.4)
σ
(
T
|
Im
⁡
P
)
=
σ
(
T
)
∩
D
(
0,1
)
,
σ
(
T
|
Im
⁡
(
I
-
P
)
)
=
σ
(
T
)
∩
(
C
∖
D
(
0,1
)
)
,
we obtain the spectral radius r
(
T
|
Im
P
)
<
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, (3.5)
‖
T
n
x
‖
≤
N
e
-
ν
n
‖
x
‖
,
∀
x
∈
Im
⁡
P
,
‖
T
n
x
‖
≥
N
-
1
e
ν
n
‖
x
‖
,
∀
x
∈
Im
⁡
(
I
-
P
)
(for the previous exposure we consulted [14, page 28]).Definition 3.2.
A semigroup{
T
(
t
)
}
t
≥
0 is hyperbolic if there exists t
0
>
0 such that T
(
t
0
) is an hyperbolic operator.Definition 3.3.
The semigroup{
T
(
t
)
}
t
≥
0 has an exponential dichotomy (or that it is exponentially dichotomic) if there exist a projection P (i.e., P
∈
ℬ
(
X
) and P
2
=
P) and the constants N
,
ν
>
0 such that  (i)
P
T
(
t
)
=
T
(
t
)
P, for all t
≥
0,
(ii)
T
(
t
)
|
Ker
P
:
Ker
P
→
Ker
P is an isomorphism, for each t
≥
0,
(iii)
∥
T
(
t
)
x
∥
≤
N
e
-
ν
t
∥
x
∥, for all t
≥
0 and x
∈
Im
P,
(iv)
∥
T
(
t
)
|
Ker
P
-
1
x
∥
≤
N
e
-
ν
t
∥
x
∥, for all t
≥
0 and x
∈
Ker
P.The first condition in the previous definition expresses equivalently thatIm
P and Ker
P are both T
(
t
)-invariant; the essence of (ii) is that {
T
(
t
)
|
Ker
P
}
t
≥
0 can be extended to a group. The next result establishes the relation between hyperbolicity and exponential dichotomy for C
0-semigroups on complex Banach spaces. For the proof we refer the reader to [14, Lemma  2.15], [20, Theorem  1.1], or alternatively [15].Proposition 3.4.
For a strongly continuous semigroup{
T
(
t
)
}
t
≥
0 acting on a complex Banach space the following statements are equivalent.  (i)
{
T
(
t
)
}
t
≥
0  is hyperbolic.
(ii)
{
T
(
t
)
}
t
≥
0  has an exponential dichotomy. Moreover, if (i) holds, then X
=
X
1
⊕
X
2 where X
1
=
Im
P, X
2
=
Im
(
I
-
P
), and
(3.6)
P
=
1
2
π
i
∫
Γ
(
λ
I
-
T
(
t
0
)
)
-
1
d
λ
is the spectral Riesz projection for T
(
t
0
) that corresponds to σ
(
T
(
t
0
)
)
∩
Γ
=
∅. Also, if  {
T
(
t
)
}
t
≥
0 has an exponential dichotomy, then σ
(
T
(
t
)
)
∩
Γ
=
∅, for every t
≥
0.IfA generates an exponentially dichotomic C
0-semigroup, then the differential equation x
̇
=
A
x has the property that the solutions x
(
·
) starting from X
1 (resp., from X
2) decay exponentially for t
>
0 (resp., for t
<
0) uniformly with respect to the initial data. As it can be seen, the exponential dichotomy concept generalizes strongly the exponential stability concept but it has a serious drawback. It forces the solution that starts from X
2 to exist for negative time, or in counterpart it forces the semigroup to be invertible on X
2. We will drop off this requirement here and extend the notion of hyperbolicity by replacing the exponential decay in negative time for the solutions starting in X
2 with an exponential blow-up in positive time. We will call the “exponential decay on X
1 and exponential blow-up on X
2” (both on positive time) behavior as  no past exponential dichotomy.Definition 3.5.
The semigroup{
T
(
t
)
}
t
≥
0 has a no past exponential dichotomy if there exist a projection P and the constants N
1
,
N
2
,
ν
>
0 such that  (i)
P
T
(
t
)
P
=
T
(
t
)
P, for all t
≥
0,
(ii)
∥
T
(
t
)
x
∥
≤
N
1
e
-
ν
t
∥
x
∥, for all t
≥
0, x
∈
Im
P,
(iii)
∥
T
(
t
)
x
∥
≥
N
2
e
ν
t
∥
x
∥, for all t
≥
0, x
∈
Ker
P.Definition 3.6.
The semigroup{
T
(
t
)
}
t
≥
0 has an ordinary dichotomy if there exist a projection P and the constants N
1
,
N
2
>
0 such that  (i)
P
T
(
t
)
P
=
T
(
t
)
P, for all t
≥
0,
(ii)
∥
T
(
t
)
x
∥
≤
N
1
∥
x
∥, for all t
≥
0, x
∈
Im
P,
(iii)
∥
T
(
t
)
x
∥
≥
N
2
∥
x
∥, for all t
≥
0, x
∈
Ker
P.Remark 3.7.
Note that ifP is one-to-one, then Im
P
=
X and thus the concept of (no past) exponential dichotomy overlaps the concept of exponential stability. Recall that {
T
(
t
)
}
t
≥
0 is said to be exponentially stable if one of the following equivalent statements is true:  (i)
there existN
,
ν
>
0 such that ∥
T
(
t
)
∥
≤
N
e
-
ν
t for all t
≥
0;
(ii)
there existt
0
>
0 such that ∥
T
(
t
0
)
∥
<
1.It is obvious that the existence of an exponential dichotomy implies the existence of a no past exponential dichotomy, but the converse is not valid as the following example points out.Example 3.8.
The followingC
0-semigroup {
S
(
t
)
}
t
≥
0
has a no past exponential dichotomy but is not exponentially dichotomic. Let A be a p
×
p matrix with real entries whose spectrum is contained in the open left-half plane and consider the right shift semigroup on L
1
(
ℝ
+
,
ℝ
), given by
(3.7)
T
(
t
)
f
(
s
)
=
{
0
,
0
≤
s
<
t
,
f
(
s
-
t
)
,
s
≥
t
.
Then S
(
t
)
=
e
t
A
⊕
e
t
T
(
t
)
,
S
(
t
)
(
x
,
f
)
=
(
e
t
A
x
,
e
t
T
(
t
)
f
) has a no past exponential dichotomy on X
=
ℝ
p
⊕
L
1
(
ℝ
+
,
ℝ
) with X
1
=
{
(
x
,
0
)
:
x
∈
ℝ
p
} and X
2
=
{
(
0
,
f
)
:
f
∈
L
1
(
ℝ
+
,
ℝ
)
}. However, the restriction of S
(
t
) on X
2 is not onto, and thus {
S
(
t
)
}
t
≥
0 is not exponentially dichotomic.Lemma 3.9.
Let{
T
(
t
)
}
t
≥
0 be an exponentially bounded semigroup. If there exist Y, a vector subspace of X, n
0
∈
ℕ
*, H
>
0, and η
>
1 such that  (i)
∥
T
(
n
)
x
∥
≥
H
∥
x
∥, for all n
∈
{
0,1
,
…
,
n
0
} and x
∈
Y,
(ii)
∥
T
(
n
+
n
0
)
x
∥
≥
η
∥
T
(
n
)
x
∥, for all n
∈
ℕ and x
∈
Y, then there exist N
,
ν
>
0 which satisfy ∥
T
(
t
)
x
∥
≥
N
e
ν
t
∥
x
∥, for all x
∈
Y and t
≥
0.Proof.
Letx
∈
Y, and n
∈
ℕ
*. Then, there exist k
∈
ℕ and r
∈
{
0,1
,
…
,
n
0
-
1
} such that n
=
k
n
0
+
r. Therefore, ∥
T
(
n
)
x
∥
=
∥
T
(
k
n
0
+
r
)
x
∥
≥
η
k
∥
T
(
r
)
x
∥
≥
H
η
k
∥
x
∥. We take ν
:
=
(
1
/
n
0
)
ln
η
>
0 to obtain
(3.8)
‖
T
(
n
)
x
‖
≥
H
e
ν
k
n
0
‖
x
‖
=
H
e
-
ν
r
e
ν
(
k
n
0
+
r
)
‖
x
‖
>
H
e
-
ν
n
0
e
ν
n
‖
x
‖
,
∀
n
∈
N
.
For t
≥
0, take n
=
[
t
]
+
1 to have M
e
ω
∥
T
(
t
)
x
∥
≥
∥
T
(
n
)
x
∥
≥
H
e
-
ν
n
0
e
ν
n
∥
x
∥. Put N
:
=
H
M
-
1
e
-
(
ν
n
0
+
ω
)
>
0 in order to write ∥
T
(
t
)
x
∥
≥
N
e
ν
t
∥
x
∥.Lemma 3.10.
Let{
T
(
t
)
}
t
≥
0 be an exponentially bounded semigroup. If there exist Y a vector subspace of X, n
0
∈
ℕ
*, H
>
0, and η
∈
(
0,1
) such that  (i)
∥
T
(
n
)
x
∥
≤
H
∥
x
∥, for all n
∈
{
0,1
,
…
,
n
0
} and x
∈
Y and
(ii)
∥
T
(
n
+
n
0
)
x
∥
≤
η
∥
T
(
n
)
x
∥, for all n
∈
ℕ and x
∈
Y, then there exist N
,
ν
>
0 which satisfy ∥
T
(
t
)
x
∥
≤
N
e
-
ν
t
∥
x
∥, for all x
∈
Y and t
≥
0.Proof.
It is analogous with the proof of Lemma3.9.Consider the autonomous inhomogeneous abstract Cauchy problem(
A
,
f
;
x
0
): (3.9)
x
̇
(
t
)
=
A
x
(
t
)
+
f
(
t
)
,
t
≥
0
,
x
(
0
)
=
x
0
.
If A generates the C
0-semigroup {
T
(
t
)
}
t
≥
0, x
∈
X and f
∈
L
loc
1
(
ℝ
+
,
X
), then the function u
:
ℝ
+
→
X given by
(3.10)
u
(
t
)
=
T
(
t
)
x
+
∫
0
t
T
(
t
-
s
)
f
(
s
)
d
s
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
′
=
A
x
+
f. The crudest expression of this method is the notion of admissibility of a pair of classes of functions both in L
loc
1
(
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
′
=
A
x.Definition 3.11.
Letℰ
,
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
(3.11)
p
f
(
n
;
x
)
=
T
(
n
)
x
+
∑
k
=
0
n
T
(
n
-
k
)
f
(
k
)
,
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
(4.1)
X
1
,
F
=
{
x
∈
X
:
(
T
(
n
)
x
)
n
∈
N
∈
F
(
X
)
}
,
X
1
=
{
x
∈
X
:
lim
⁡
t
→
∞
T
(
t
)
x
=
0
}
,
which are obviously vector subspaces of X.Hypothesis 1.
The vector subspaceX
1
,
ℱ is closed and admits a closed complement; that is, there exists X
2
,
ℱ, a closed vector subspace, such that
(4.2)
X
=
X
1
,
F
⊕
X
2
,
F
.
We denote byP
1 the projection onto X
1
,
ℱ along X
2
,
ℱ and set P
2
=
I
-
P
1 (we will prove that in the case of a no past exponential dichotomy for {
T
(
t
)
}
t
≥
0, X
1
,
ℱ coincides always with X
1 and thus the ℱ-independent notation for projectors is consistent).Remark 4.1.
We have thatT
(
t
)
X
1
,
ℱ
⊂
X
1
,
ℱ and T
(
t
)
x
≠
0, for each x
∈
X
2
,
ℱ
∖
{
0
} and t
≥
0.Proof.
Ifx
∈
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 existt
≥
0 and x
∈
X
2
,
ℱ
∖
{
0
} such that T
(
t
)
x
=
0. Then, T
(
n
)
x
=
T
(
n
-
t
)
T
(
t
)
x
=
0 for every n
∈
ℕ, n
≥
t and thus (
T
(
n
)
x
)
n
∈
ℕ
∈
ℱ
(
X
). It follows that x
∈
X
1
,
ℱ, which is not possible since x
∈
X
2
,
ℱ
∖
{
0
}. Thus, T
(
t
)
|
X
2
,
ℱ is one-to-one, for all t
≥
0.Proposition 4.2.
If the pair(
ℰ
,
ℱ
) is admissible to the semigroup {
T
(
t
)
}
t
≥
0, then for each f
∈
ℰ
(
X
), there exists unique x
f
∈
X
2
,
ℱ such that p
f
(
·
;
x
f
)
∈
ℱ
(
X
).Proof.
Letf
∈
ℰ
(
X
) and x
∈
X from Definition 3.11. For y
=
x
-
P
1
x
=
P
2
x we have that y
∈
X
2
,
ℱ and p
f
(
n
;
y
)
=
p
f
(
n
;
x
)
-
T
(
n
)
P
1
x. Since p
f
(
·
;
x
)
∈
ℱ
(
X
) and (
T
(
n
)
P
1
x
)
∈
ℱ
(
X
), it follows that p
f
(
·
;
y
)
∈
ℱ
(
X
).
To prove the uniqueness ofy, 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 vectory
∈
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
(4.3)
‖
x
f
‖
≤
K
‖
f
‖
E
(
X
)
,
‖
p
f
(
⋅
;
x
f
)
‖
F
(
X
)
≤
K
‖
f
‖
E
(
X
)
,
for all f
∈
ℰ
(
X
).Proof.
We define the operator(4.4)
U
:
E
(
X
)
⟶
X
2
,
F
⊕
F
(
X
)
,
U
f
=
(
x
f
,
p
f
(
⋅
;
x
f
)
)
.
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
(4.5)
‖
U
f
n
-
(
y
,
g
)
‖
X
2
,
F
⊕
F
(
X
)
→
n
→
∞
0
,
where f
∈
ℰ
(
X
), y
∈
X
2
,
ℱ, and g
∈
ℱ
(
X
). For each n
∈
ℕ, we take x
n
=
x
f
n
∈
X
2
,
ℱ and u
n
=
p
f
n
(
·
;
x
n
)
∈
ℱ
(
X
). We have that ∥
x
n
-
y
∥
→
n
→
∞
0 and ∥
u
n
-
g
∥
ℱ
(
X
)
→
n
→
∞
0.
Since(4.6)
‖
u
n
(
m
)
-
g
(
m
)
‖
≤
1
β
F
(
0
)
‖
u
n
-
g
‖
F
(
X
)
,
we have that lim
n
→
∞
u
n
(
m
)
=
g
(
k
), for all m
∈
ℕ. On the other hand,
(4.7)
‖
u
n
(
m
)
-
p
f
(
m
;
y
)
‖
=
‖
T
(
m
)
(
x
n
-
y
)
+
∑
j
=
0
m
T
(
m
-
j
)
(
f
n
(
j
)
-
f
(
j
)
)
‖
≤
‖
T
(
m
)
‖
‖
x
n
-
y
‖
+
∑
j
=
0
m
‖
T
(
m
-
j
)
‖
‖
f
n
(
j
)
-
f
(
j
)
‖
,
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 𝒰
f
=
(
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
(4.8)
‖
x
f
‖
+
‖
p
f
(
⋅
;
x
f
)
‖
F
(
X
)
=
‖
U
f
‖
X
2
,
F
⊕
F
(
X
)
≤
K
‖
f
‖
E
(
X
)
,
and the proof is complete.A simple and useful evaluation that results from aProposition4.3 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.
Letℰ and ℱ be sequence Schäffer spaces such that ℓ
1
≠
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
=
X
1.Proof (Part I).
The exponential decay of{
T
(
t
)
}
t
≥
0 on X
1
,
ℱ.
Letx
∈
X
1
,
ℱ and consider the sequence
(4.9)
f
:
N
⟶
X
,
f
(
n
)
=
δ
0
(
n
)
x
,
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
(4.10)
‖
T
(
t
)
x
‖
≤
M
e
ω
(
t
-
n
)
‖
T
(
n
)
x
‖
≤
M
e
ω
K
β
E
(
0
)
β
F
(
0
)
‖
x
‖
.
Since the constant C
1
:
=
M
e
ω
K
(
β
ℰ
(
0
)
/
β
F
(
0
)
) does not depend on x, we can write down
(4.11)
‖
T
(
t
)
x
‖
≤
C
1
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
1
,
F
.
For n
,
m
∈
ℕ and x
∈
X
1
,
ℱ, we evaluate
(4.12)
‖
T
(
n
)
x
‖
∑
j
=
0
n
δ
j
(
m
)
=
∑
j
=
0
n
‖
T
(
n
-
j
)
T
(
j
)
x
‖
δ
j
(
m
)
≤
C
1
∑
j
=
0
n
‖
T
(
j
)
x
‖
δ
j
(
m
)
≤
C
1
∑
j
=
0
∞
‖
T
(
j
)
x
‖
δ
j
(
m
)
=
C
1
‖
p
f
(
m
;
0
)
‖
,
which implies ∥
T
(
n
)
x
∥
∑
j
=
0
n
δ
j
≤
C
1
∥
p
f
(
·
;
0
)
∥ and therefore
(4.13)
‖
T
(
n
)
x
‖
∥
χ
{
0,1
,
…
,
n
}
∥
F
≤
C
1
∥
p
f
(
⋅
;
0
)
∥
F
(
X
)
.
By Proposition 4.3 we get that
(4.14)
‖
T
(
n
)
x
‖
≤
C
1
K
β
E
(
0
)
β
F
(
n
)
‖
x
‖
,
∀
n
∈
N
,
x
∈
X
1
,
F
.
Ifℓ
0
∞
⊄
ℱ, then β
ℱ is not bounded and therefore there exists n
0
∈
ℕ
* such that η
:
=
C
1
K
β
ℰ
(
0
)
/
β
ℱ
(
n
0
)
<
1 and ∥
T
(
n
0
)
x
∥
≤
η
∥
x
∥, for all x
∈
X
1
,
ℱ.
Ifℓ
0
∞
⊂
ℱ, then ℰ
≠
ℓ
1. From Proposition 2.8, it follows that there exists h
∈
ℰ
∖
ℓ
1. Consider
(4.15)
γ
:
N
⟶
R
,
γ
(
n
)
=
∑
j
=
0
n
|
h
(
j
)
|
,
which is nondecreasing and lim
n
→
∞
γ
(
n
)
=
∞. For n
∈
ℕ and x
∈
X
1
,
ℱ, the sequence
(4.16)
g
:
N
⟶
X
,
g
(
m
)
=
χ
{
0,1
,
…
,
n
}
(
m
)
|
h
(
m
)
|
T
(
m
)
x
has finite support, and thus g
∈
ℰ
(
X
). Observing that ∥
g
(
m
)
∥
≤
C
1
∥
x
∥
|
h
(
m
)
| for every m
∈
ℕ, we are led to the evaluation ∥
g
∥
ℰ
(
X
)
≤
L
∥
h
∥
ℰ
∥
x
∥. Also,
(4.17)
p
g
(
m
;
0
)
=
∑
j
=
0
m
T
(
m
-
j
)
g
(
j
)
=
∑
j
=
0
m
T
(
m
-
j
)
T
(
j
)
x
|
h
(
j
)
|
χ
{
0,1
,
…
,
n
}
(
j
)
=
(
∑
j
=
0
m
χ
{
0,1
,
…
,
n
}
(
j
)
|
h
(
j
)
|
)
T
(
m
)
x
,
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
(4.18)
γ
(
n
)
‖
T
(
n
)
x
‖
≤
K
C
1
β
F
(
0
)
‖
h
‖
E
‖
x
‖
.
It follows that there exists n
0
∈
ℕ
* such that η
:
=
(
K
C
1
∥
h
∥
ℰ
/
β
ℱ
(
0
)
γ
(
n
0
)
)
<
1 and ∥
T
(
n
0
)
x
∥
≤
η
∥
x
∥, for all x
∈
X
1
,
ℱ.
In both cases, we obtained the existence of somen
0
∈
ℕ
* and some constant η
∈
(
0,1
) such that ∥
T
(
n
0
)
x
∥
≤
η
∥
x
∥ for all x
∈
X
1
,
ℱ. Therefore, the semigroup {
T
(
t
)
|
X
1
,
ℱ
}
t
≥
0 is exponentially stable (see Remark 3.7). Subsequently, there exist N
1
,
ν
1
>
0 such that
(4.19)
‖
T
(
t
)
x
‖
≤
N
1
e
-
ν
1
t
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
1
,
F
.Proof (Part II).
The exponential blow-up of{
T
(
t
)
}
t
≥
0 on X
2
,
ℱ.
Letn
∈
ℕ, n
0
∈
ℕ
* and x
∈
X
2
,
ℱ
∖
{
0
}, and consider the sequence
(4.20)
f
:
N
⟶
X
,
f
(
m
)
=
δ
n
+
n
0
(
m
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
.
We have that f
∈
ℰ
(
X
) with ∥
f
∥
ℰ
(
X
)
=
β
ℰ
(
0
). On the one hand,
(4.21)
-
∑
j
=
m
+
1
∞
δ
n
+
n
0
(
j
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
=
-
∑
j
=
0
∞
δ
n
+
n
0
(
j
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
+
∑
j
=
0
m
δ
n
+
n
0
(
k
)
T
(
m
-
j
)
T
(
j
)
x
‖
T
(
n
+
n
0
)
x
‖
=
T
(
m
)
(
-
x
‖
T
(
n
+
n
0
)
x
‖
)
+
∑
j
=
0
m
T
(
m
-
j
)
f
(
j
)
=
p
f
(
m
;
y
)
,
for all m
∈
ℕ (where y
=
-
x
/
∥
T
(
n
+
n
0
)
x
∥
∈
X
2
,
ℱ), while on the other hand
(4.22)
-
∑
j
=
m
+
1
∞
δ
n
+
n
0
(
j
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
=
{
-
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
,
m
<
n
+
n
0
,
0
,
m
≥
n
+
n
0
,
which implies that p
f
(
·
;
y
)
∈
ℱ
(
X
). Then, ∥
p
f
(
m
;
y
)
∥
≤
K
/
(
β
ℱ
(
0
)
)
∥
f
∥
ℰ
(
X
) and
(4.23)
‖
T
(
m
)
x
‖
≤
K
β
E
(
0
)
β
F
(
0
)
‖
T
(
n
+
n
0
)
x
‖
,
∀
0
≤
m
<
n
+
n
0
.
Taking now m
=
n
=
0 we can write down
(4.24)
‖
T
(
n
0
)
x
‖
≥
K
-
1
β
F
(
0
)
β
E
(
0
)
‖
x
‖
,
∀
n
0
∈
N
*
,
x
∈
X
2
,
F
.
For t
≥
0, we put n
0
=
[
t
]
+
1 to evaluate M
e
ω
∥
T
(
t
)
x
∥
≥
∥
T
(
n
0
)
x
∥
≥
K
-
1
(
β
ℱ
(
0
)
/
β
ℰ
(
0
)
)
∥
x
∥. Denoting C
2
:
=
(
K
M
e
ω
)
-
1
β
ℱ
(
0
)
/
β
ℰ
(
0
) we have that
(4.25)
‖
T
(
t
)
x
‖
≥
C
2
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
2
,
F
.
Also, from (4.23), taking C
′
:
=
1
+
K
(
β
ℰ
(
0
)
/
β
ℱ
(
0
)
) we have that
(4.26)
C
′
‖
T
(
h
)
x
‖
≥
‖
T
(
n
)
x
‖
,
∀
h
≥
n
,
x
∈
X
2
,
F
.
Since n
,
n
0, and x are randomly taken (see the beginning of Part II), we have that
(4.27)
‖
T
(
n
)
x
‖
‖
T
(
n
+
n
0
)
x
‖
∑
j
=
n
n
+
n
0
-
1
δ
j
(
m
)
≤
C
′
∑
j
=
n
n
+
n
0
-
1
‖
T
(
j
)
x
‖
‖
T
(
n
+
n
0
)
x
‖
δ
j
(
m
)
=
C
′
∑
j
=
n
n
+
n
0
-
1
‖
p
f
(
j
;
y
)
‖
δ
j
(
m
)
≤
C
′
‖
p
f
(
m
;
y
)
‖
,
∀
m
∈
N
,
and therefore ∥
T
(
n
)
x
∥
/
∥
T
(
n
+
n
0
)
x
∥
χ
{
n
,
n
+
1
,
…
,
n
+
n
0
-
1
}
≤
C
′
∥
p
f
(
·
;
y
)
∥. It follows that
(4.28)
‖
T
(
n
)
x
‖
‖
T
(
n
+
n
0
)
x
‖
‖
R
n
χ
{
0,1
,
…
,
n
0
-
1
}
‖
F
≤
C
′
‖
‖
p
f
(
⋅
;
y
)
‖
‖
F
,
or equivalently,
(4.29)
‖
T
(
n
)
x
‖
‖
T
(
n
+
n
0
)
x
‖
β
F
(
n
0
-
1
)
≤
C
′
‖
p
f
(
⋅
;
y
)
‖
F
(
X
).,
Using the fact that (4.29)
∥
p
f
(
⋅
;
y
)
∥
F
(
X
)
≤
K
∥
f
∥
E
(
X
), we deduce that
(4.29)
‖
T
(
n
+
n
0
)
x
‖
≥
β
F
(
n
0
-
1
)
C
′
K
β
E
(
0
)
‖
T
(
n
)
x
‖
,
∀
n
∈
N
,
n
0
∈
N
*
,
x
∈
X
2
,
F
.
Ifℓ
0
∞
⊄
ℱ, then β
ℱ is not bounded and therefore there exists n
0
∈
ℕ
* such that η
:
=
(
β
ℱ
(
n
0
-
1
)
)
/
(
C
′
K
β
ℰ
(
0
)
)
>
1 and ∥
T
(
n
+
n
0
)
x
∥
≥
η
∥
T
(
n
)
x
∥ for all n
∈
ℕ and x
∈
X
2
,
ℱ.
Ifℓ
0
∞
⊂
ℱ, then ℰ
≠
ℓ
1 and therefore there exists h
∈
ℰ
∖
ℓ
1. Consider γ as in (4.15), and for n
,
n
0
∈
ℕ and x
∈
X
2
,
ℱ
∖
{
0
} we define
(4.31)
g
:
N
⟶
X
,
g
(
m
)
=
∑
j
=
0
n
0
δ
n
+
j
(
m
)
|
h
(
j
)
|
T
(
n
+
j
)
x
‖
T
(
n
+
n
0
)
x
‖
.
Since ∥
g
(
m
)
∥
≤
C
′
χ
{
n
,
n
+
1
,
…
,
n
+
n
0
}
(
m
)
|
(
R
n
h
)
(
m
)
|
≤
C
′
|
(
R
n
h
)
(
m
)
| for each m
∈
ℕ, we have that g
∈
ℰ
(
X
) with ∥
g
∥
ℰ
(
X
)
≤
C
′
∥
R
n
h
∥
ℰ
=
C
′
∥
h
∥
ℰ. Note that g
(
m
)
=
χ
{
n
,
n
+
1
,
…
,
n
+
n
0
}
(
m
)
|
R
n
h
(
m
)
|
T
(
m
)
x
/
∥
T
(
n
+
n
0
)
x
∥, for all m
∈
ℕ. On the one hand,
(4.32)
-
∑
j
=
m
+
1
∞
χ
{
n
,
…
,
n
+
n
0
}
(
j
)
|
R
n
h
(
j
)
|
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
=
-
∑
j
=
0
∞
χ
{
n
,
…
,
n
+
n
0
}
(
j
)
|
R
n
h
(
j
)
|
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
+
+
∑
j
=
0
m
T
(
m
-
j
)
χ
{
n
,
…
,
n
+
n
0
}
(
j
)
|
R
n
h
(
j
)
|
T
(
j
)
x
‖
T
(
n
+
n
0
)
x
‖
=
T
(
m
)
(
γ
(
n
0
)
-
x
‖
T
(
n
+
n
0
)
x
‖
)
+
∑
j
=
0
m
T
(
m
-
j
)
g
(
j
)
=
p
g
(
m
;
z
)
,
for all m
∈
ℕ (where z
:
=
γ
(
n
0
)
(
-
x
/
∥
T
(
n
+
n
0
)
x
∥
)
∈
X
2
,
ℱ), while on the other hand
(4.33)
-
∑
j
=
m
+
1
∞
χ
{
n
,
…
,
n
+
n
0
}
(
j
)
|
R
n
h
(
j
)
|
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
=
{
-
γ
(
n
0
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
,
m
<
n
,
-
(
γ
(
n
0
)
-
γ
(
m
-
n
)
)
T
(
m
)
x
‖
T
(
n
+
n
0
)
x
‖
,
n
≤
m
<
n
+
n
0
,
0
,
m
≥
n
+
n
0
.
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
(4.34)
(
γ
(
n
0
)
-
|
h
(
0
)
|
)
∥
T
(
n
)
x
∥
∥
T
(
n
+
n
0
)
x
∥
≤
K
C
′
β
F
(
0
)
∥
h
∥
E
.
Thus, ∥
T
(
n
+
n
0
)
x
∥
≥
(
β
ℱ
(
0
)
/
K
C
′
∥
h
∥
ℰ
)
(
γ
(
n
0
)
-
|
h
(
0
)
|
)
∥
T
(
n
)
x
∥, for all n
,
n
0
∈
ℕ and x
∈
X
2
,
ℱ. Then, there exist n
0
∈
ℕ
* and η
>
1 such that
(4.35)
‖
T
(
n
+
n
0
)
x
‖
≥
η
‖
T
(
n
)
x
‖
,
∀
n
∈
N
,
x
∈
X
2
,
F
.
In both cases, we obtained the existence of somen
0
∈
ℕ
* and some constant η
>
1 satisfying the condition (ii) from Lemma 3.9, while the first condition is assured by (4.25). Subsequently, there exist N
2
,
ν
2
>
0 such that
(4.36)
‖
T
(
t
)
x
‖
≥
N
2
e
ν
2
t
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
2
,
F
.
If we takeν
:
=
min
{
ν
1
,
ν
2
}
>
0, all the conditions guaranteeing the existence of a no past exponential dichotomy are met.Proof (Part III).
We prove thatX
1
,
ℱ
=
X
1, no matter how we choose the sequence Schäffer space ℱ. If x
∈
X
1
,
ℱ, then ∥
T
(
t
)
x
∥
≤
N
1
e
-
ν
1
t
∥
x
∥, for all t
≥
0, which implies x
∈
X
1.
Conversely, letx
∈
X
1, u
∈
X
1
,
ℱ, and v
∈
X
2
,
ℱ such that x
=
u
+
v. For every n
∈
ℕ, we have that
(4.37)
‖
T
(
n
)
x
‖
=
‖
T
(
n
)
v
-
(
-
T
(
n
)
u
)
‖
≥
‖
T
(
n
)
v
‖
-
‖
T
(
n
)
u
‖
≥
N
2
e
ν
2
n
‖
v
‖
-
N
1
e
-
ν
1
n
‖
u
‖
.
If we suppose that v
≠
0, then ∥
T
(
n
)
x
∥
→
n
→
∞
∞ contradicting the fact that x
∈
X
1. It follows immediately that v
=
0 and x
=
u
∈
X
1
,
ℱ.Remark 4.6.
As we pointed out in the introduction, there is an extensive literature on the connection betweenadmissibility 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 C
0-semigroup {
T
(
t
)
}
t
≥
0, when we assume a priori that the kernel of the splitting projection is T
(
t
)-invariant and T
(
t
)
|
Ker
P are invertible. For details we refer the reader to [21]. We try to extend this line of results in two directions. First, we do not assume a  priori that T
(
t
)
|
Ker
P is invertible (we do not even assume that Ker
P is T
(
t
)-invariant) and still we succeed to prove that the admissibility of any pair of sequence Schäffer spaces implies the existence of a no past exponential dichotomy. Secondly, it is worth to note that the class of sequence Schäffer spaces is extremely reachable (see e.g., Examples 2.3 and 2.4) and this fact allows the reader to choose the test sequences” in various ways and in the same time it does not force the “output” or “nice 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 ℓ
1
≠
ℰ or ℓ
0
∞
⊄
ℱ, then the subspace X
1
,
ℱ (which induces the no past exponential dichotomy) is actually the regular stable subspace X
1. If we would impose in addition that the complement of X
1 (denoted by X
2) is also T
(
t
)-invariant, then the aforementioned admissibility condition would imply that {
T
(
t
)
}
t
≥
0 extends automatically to a C
0-group on X
2, and thus we would get hyperbolicity for {
T
(
t
)
}
t
≥
0 (see Theorem 4.12 below). Therefore, we can conclude that “admissibility” converts to “no past exponential dichotomy”“admissibility” and “T
(
t
)-invariance of X
2” converts to “hyperbolicity”.The next example shows that the condition “E
≠
ℓ
1 or ℓ
0
∞
⊄
ℱ” in the statement of Theorem 4.5 is essential.Example 4.7.
LetX
=
ℝ and consider the semigroup {
T
(
t
)
}
t
≥
0 by T
(
t
)
=
I
ℝ for all t
≥
0. If f
∈
ℓ
1, then there exists x
=
-
∑
k
=
0
∞
f
(
k
)
∈
ℝ (the series being absolutely convergent) such that p
f
(
·
;
x
)
∈
ℓ
0
∞
(
X
). Therefore, the pair (
ℓ
1
,
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 spaceX
1
,
ℱ induces a no past exponential dichotomy for {
T
(
t
)
}
t
≥
0—an exponentially bounded semigroup—and that X
1
,
ℱ is actually X
1. 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: (
B
1
,
D
1
) is said to be stronger than
(
B
,
D
) (or the later pair is weaker than the former) if B
↪
B
1 and D
1
↪
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 (
ℓ
1
,
ℓ
0
∞
) is stronger than (
ℰ
,
ℱ
) if ℰ
⊂
ℓ
1 and ℓ
0
∞
⊂
ℱ, or equivalently (considering Proposition 2.8), ℰ
=
ℓ
1 and ℓ
0
∞
⊂
ℱ. Therefore, the hypothesis “E
≠
ℓ
1 or ℓ
0
∞
⊄
ℱ” can be replaced by, “the pair (
ℓ
1
,
ℓ
0
∞
) is not stronger than the pair (
ℰ
,
ℱ
)”.
Dropping off the restriction “ℓ
1
≠
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
)
|
X
1
,
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 C
1
,
C
2
>
0 such that
(4.38)
‖
T
(
t
)
x
‖
≤
C
1
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
1
,
F
,
‖
T
(
t
)
x
‖
≥
C
2
‖
x
‖
,
∀
t
≥
0
,
x
∈
X
2
,
F
.Proof.
Note that in the proof of Theorem4.5, we do not use the hypothesis “E
≠
ℓ
1 or ℓ
0
∞
⊄
ℱ” to prove the relations (4.11) and (4.25).Corollary 4.10.
Letp
,
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 Proposition2.10 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.
Letℓ
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 Theorem4.5 and Example 2.4.
With Theorem4.12 we prove that if we impose the T
(
t
)-invariance of X
2, we can deduce the invertibility of T
(
t
)
|
X
2, 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 X
2 is invariant to all operators T
(
t
). However, we can prove that {
T
(
t
)
|
X
2
}
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.
Letℰ and ℱ be two sequence Schäffer spaces such that ℓ
1
≠
ℰ or ℓ
0
∞
⊄
ℱ and let {
T
(
t
)
}
t
≥
0 be an exponentially bounded semigroup. If(i)
there existst
0
>
0 such that P
2
T
(
t
)
P
2
=
T
(
t
)
P
2 for all t
∈
[
0
,
t
0
],
(ii)
(
ℰ
,
ℱ
) is admissible to T, then T
(
t
)
:
X
2
→
X
2 is invertible for each t
≥
0. Therefore, {
T
(
t
)
}
t
≥
0 has an exponential dichotomy.Proof.
Lett
>
t
0 and consider n
∈
ℕ and δ
∈
[
0
,
t
0
) such that t
=
n
t
0
+
δ. If x
∈
X
2, we have that T
(
t
)
x
=
T
(
δ
)
T
(
t
0
)
n
x
∈
X
2. Therefore, the condition (i) assures that the operator T
(
t
)
:
X
2
→
X
2 is well defined (with respect to the range) for each t
≥
0 and from Theorem 4.5 we have that {
T
(
t
)
}
t
≥
0 has a no past exponential dichotomy.
Lett
≥
0, n
0
=
[
t
]
+
1, y
∈
X
2 and consider the sequence
(4.39)
f
:
N
⟶
X
,
f
(
n
)
=
-
δ
n
0
(
n
)
T
(
n
0
-
t
)
y
,
which belongs to ℰ
(
X
) with ∥
f
∥
ℰ
(
X
)
=
β
ℰ
(
0
)
∥
T
(
n
0
-
t
)
y
∥. Then, according to Proposition 4.2, there exists a unique x
∈
X
2 such that p
f
(
·
;
x
)
∈
ℱ
(
X
). Since
(4.40)
p
f
(
n
;
x
)
=
T
(
n
)
x
-
∑
k
=
0
n
δ
n
0
(
k
)
T
(
n
-
k
)
T
(
n
0
-
t
)
y
=
T
(
n
)
x
-
T
(
n
-
n
0
)
T
(
n
0
-
t
)
y
=
T
(
n
-
t
)
T
(
t
)
x
-
T
(
n
-
t
)
y
=
T
(
n
-
t
)
(
T
(
t
)
x
-
y
)
,
for all n
∈
ℕ, n
≥
n
0, we have that T
(
t
)
x
,
y
∈
X
2 and
(4.41)
‖
p
f
(
n
;
x
)
‖
=
‖
T
(
n
-
t
)
(
T
(
t
)
x
-
y
)
‖
≥
N
2
e
ν
2
(
n
-
t
)
‖
T
(
t
)
x
-
y
‖
,
∀
n
≥
n
0
.
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
)
|
X
2 is onto, and since the one-to-one property was already proved in the Remark 4.1, we get the invertibility.
Since only the property of invertibility of operators onX
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 Theorem4.12. 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.
Ifℰ and ℱ are two sequence Schäffer spaces such that L
ℰ
⊂
ℱ, then ℰ
⊂
ℱ and ∥
L
f
∥
ℱ
≤
∥
f
∥
ℱ for all f
∈
ℰ.Proof.
Letf
∈
ℰ. Then, R
f
∈
ℰ and therefore f
=
L
(
R
f
)
∈
ℱ and L
f
∈
ℱ. Since, |
R
L
f
(
m
)
|
≤
|
f
(
m
)
| for each m
∈
ℕ, we deduce that
(4.42)
‖
L
f
‖
F
=
‖
R
L
f
‖
F
≤
‖
f
‖
F
.Theorem 4.14.
Letℰ 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 existN
,
ν
>
0 such that
(4.43)
‖
T
(
t
)
P
1
x
‖
≤
N
e
-
ν
t
‖
P
1
x
‖
,
‖
T
(
t
)
|
Im
⁡
P
2
-
1
P
2
x
‖
≤
N
e
-
ν
t
‖
P
2
x
‖
,
for all t
≥
0 and x
∈
X.
Letf
∈
ℰ
(
X
) and consider the sequence
(4.44)
φ
:
N
⟶
X
,
φ
(
n
)
=
∑
k
=
0
n
T
(
n
-
k
)
P
1
f
(
k
)
-
∑
k
=
n
+
1
∞
T
(
k
-
n
)
-
1
P
2
f
(
k
)
,
which is well defined, since f
∈
ℓ
∞
(
X
) and
(4.45)
∑
k
=
n
+
1
∞
‖
T
(
k
-
n
)
-
1
P
2
f
(
k
)
‖
≤
N
‖
P
2
‖
∑
k
=
n
+
1
∞
e
-
ν
(
k
-
n
)
‖
f
(
k
)
‖
≤
N
‖
P
2
‖
‖
f
‖
∞
1
-
e
-
ν
,
∀
n
∈
N
.
For any n
∈
ℕ, we have the following evaluation:
(4.46)
‖
φ
(
n
)
‖
≤
N
‖
P
1
‖
∑
k
=
0
n
e
-
ν
(
n
-
k
)
‖
f
(
k
)
‖
+
N
‖
P
2
‖
∑
k
=
n
+
1
∞
e
-
ν
(
k
-
n
)
‖
f
(
k
)
‖
,
≤
N
(
‖
P
1
‖
+
‖
P
2
‖
)
(
∑
k
=
0
n
e
-
ν
k
‖
R
k
f
‖
(
n
)
+
∑
k
=
1
∞
e
-
ν
k
‖
L
k
f
‖
(
n
)
)
.
From the hypothesis and Remark 4.13 we have that f
,
R
k
f
,
L
k
f
∈
ℱ
(
X
) with ∥
L
k
f
∥
ℱ
(
X
)
≤
∥
f
∥
ℱ
(
X
) for all k
∈
ℕ. Therefore,
(4.47)
∑
k
=
1
∞
‖
e
-
ν
k
‖
L
k
f
‖
‖
F
≤
∑
k
=
1
∞
e
-
ν
k
‖
f
‖
F
(
X
)
=
e
-
ν
1
-
e
-
ν
‖
f
‖
F
(
X
)
,
∑
k
=
0
∞
‖
e
-
ν
k
‖
R
k
f
‖
‖
F
=
∑
k
=
0
∞
e
-
ν
k
‖
f
‖
F
(
X
)
=
1
1
-
e
-
ν
‖
f
‖
F
(
X
)
,
and thus g
:
=
∑
k
=
0
n
e
-
ν
k
∥
R
k
f
∥
+
∑
k
=
1
∞
e
-
ν
k
∥
L
k
f
∥ exists as an element in ℱ.
From (4.46), denoting C
:
=
N
(
∥
P
1
∥
+
∥
P
2
∥
) we have that ∥
φ
(
n
)
∥
≤
C
g
(
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
)
-
1
x
=
T
(
n
)
T
(
k
)
-
1
x for all x
∈
X
2.
Then,(4.48)
φ
(
n
)
=
∑
k
=
0
n
T
(
n
-
k
)
P
1
f
(
k
)
-
∑
k
=
0
∞
T
(
n
)
T
(
k
)
-
1
P
2
f
(
k
)
+
∑
k
=
0
n
T
(
n
-
k
)
T
(
k
)
T
(
k
)
-
1
P
2
f
(
k
)
=
T
(
n
)
(
-
∑
k
=
0
∞
T
(
k
)
-
1
P
2
f
(
k
)
)
+
∑
k
=
0
n
T
(
n
-
k
)
(
P
1
f
(
k
)
+
P
2
f
(
k
)
)
=
T
(
n
)
x
+
∑
k
=
0
n
T
(
n
-
k
)
f
(
k
)
=
p
f
(
n
;
x
)
,
∀
n
∈
N
,
where x
=
-
∑
k
=
0
∞
T
(
k
)
-
1
P
2
f
(
k
)
∈
X
2 (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 conditionL
(
ℰ
∪
ℱ
)
⊂
ℱ and the proof of the above theorem. The condition is in fact equivalent with L
ℱ
⊂
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 L
ℰ
⊂
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 existst
0
>
0 such that P
2
T
(
t
)
P
2
=
T
(
t
)
P
2 for all t
∈
[
0
,
t
0
];
(ii)
ℰ
⊂
ℱ;
(iii)
L
ℰ
⊂
ℰ or L
ℱ
⊂
ℱ;
(iv)
ℓ
1
≠
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.
Thenecessity follows from Theorem 4.14 and Remark 4.15, while the sufficiency is proved by Theorems 4.5 and 4.12.Example 4.17.
Consider the semigroup acting on(
ℝ
,
|
·
|
) defined by T
(
t
)
x
=
e
-
t
x. Clearly, {
T
(
t
)
}
t
≥
0 defines a C
0-semigroup with ∥
T
(
t
)
x
∥
=
e
-
t
|
x
| for all t
≥
0 and x
∈
X. Therefore, {
T
(
t
)
}
t
≥
0 is exponentially stable (thus it is exponentially dichotomic).
Consider the sequencef
:
ℕ
→
ℝ
f
(
n
)
=
1
/
(
n
+
1
). We have that f
∈
ℓ
2
∖
ℓ
1, and for any x
∈
ℝ,
(4.49)
p
f
(
n
;
x
)
=
T
(
n
)
x
+
∑
k
=
0
n
T
(
n
-
k
)
f
(
k
)
=
e
-
n
x
+
∑
k
=
0
n
e
-
(
n
-
k
)
1
k
+
1
.
It follows that |
p
f
(
n
;
x
)
|
≥
1
/
(
n
+
1
)
-
e
-
n
|
x
|, for all n
∈
ℕ. Then,
(4.50)
∑
n
=
0
∞
|
p
f
(
n
;
x
)
|
≥
∑
n
=
0
∞
1
n
+
1
-
|
x
|
1
-
e
-
1
.
We obtained that p
f
(
·
;
x
)
∉
ℓ
1 for any x
∈
X. Thus, the pair (
ℓ
2
,
ℓ
1
) is not admissible to {
T
(
t
)
}
t
≥
0.
This example shows that the conditionℰ
⊂
ℱ in Theorem 4.14 cannot be dropped. If the semigroup is exponentially stable, then X
2
=
{
0
}, and therefore, the invariance condition of either ℰ or ℱ under the left shift is not necessary.Corollary 4.18.
Letℰ 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 withP
1
=
I. With this remark, observe that the expression of φ in (4.44) reduces to
(4.51)
φ
(
n
)
=
∑
k
=
0
n
T
(
n
-
k
)
f
(
k
)
,
and, respectively, the evaluation from (4.46) becomes
(4.52)
‖
φ
(
n
)
‖
≤
N
∑
k
=
0
n
e
-
ν
k
‖
R
k
f
‖
(
n
)
≤
N
∑
k
=
0
∞
e
-
ν
k
‖
R
k
f
‖
(
n
)
.
The sequence g
:
=
∑
k
=
0
∞
e
-
ν
k
∥
R
k
f
∥ exists in ℰ and ∥
φ
(
n
)
∥
≤
g
(
n
) for all n
∈
ℕ. Therefore, φ
∈
ℰ, and since φ
=
p
f
(
·
;
0
), ℰ
⊂
ℱ, we deduce that there exists x
∈
X such that p
f
(
·
;
0
)
∈
ℱ
(
X
).Corollary 4.19.
Letp
,
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 spacesℓ
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.
Letℓ
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 Theorem4.14 and Corollary 4.11, since any ℓ
Φ is invariant under the left shift.



---

*Source: 527647-2012-01-26.xml*