Let be a real or a complex Banach space. In this section, we will present a complete study concerning the characterization of uniform exponential trichotomy using a special solvability of an associated integral equation. We introduce a new and natural admissibility concept and we show that the trichotomic behavior of skewproduct flows can be studied in the most general case, without any additional assumptions.
Notations
Let , which is a Banach space with respect to the norm . We consider the spaces , and let . Then and are closed linear subspaces of . Let be the space of all continuous functions with compact support and .
Let and let be the linear space of all Bochner measurable functions with the property that , which is a Banach space with respect to the norm
Let be a metric space and let be a skewproduct flow. For every we consider the integral equation
with and .
Definition 3.1.
The pair is said to beuniformly admissible for the skewproduct flow if there are and such that for every the following properties hold:

(i)
for every there are and such that the pairs and satisfy ( );

(ii)
if and are such that the pair satisfies ( ), then ;

(iii)
if is such that , for all and has the property that the pair satisfies ( ), then .
Remark 3.2.
In the above admissibility concept, the input space is a minimal one, because all the test functions belong to the space .
In what follows we will establish the connections between the admissibility and the existence of uniform exponential trichotomy.
The first main result of this paper is as follows.
Theorem 3.3.
Let be a skewproduct flow on . If the pair , is uniformly admissible for , then is uniformly exponentially trichotomic.
Proof.
We prove that the pair is uniformly admissible for .
Indeed, let and be given by Definition 3.1. We consider a continuous function with the support contained in and
Let be such that , for all . Since there is such that
Let .
Step 1.
Let . We consider the function
Then is continuous and
Since there is such that . Then, from (3.7) it follows that , so . According to our hypothesis it follows that there are and such that the pairs and satisfy ( ).
Then, for every we obtain that
Let
From (3.8) we have that
so the pair satisfies ( ). Moreover, since and , we deduce that . Since the pair satisfies ( ) we obtain that
Taking
we analogously obtain that and the pair satisfies ( ).
Step 2.
Let and let be such that the pair satisfies ( ). We consider the functions , given by
Since we have that . Observing that for every
we deduce that is continuous. Moreover, since and , from
we obtain that .
Let . We prove that
We set . If , then, taking into account the way how was defined, the relation (3.16) obviously holds. If then there is , such that . Then, we deduce that
If then and
If then
From relations (3.18) and (3.19) we have that
Then, from (3.17) and (3.20) it follows that
Since the pair satisfies ( ) we have that
From (3.21) and (3.22) we obtain that the relation (3.16) holds for all . Since was arbitrary we deduce that the pair satisfies ( ). Then according to our hypothesis we have that
In addition, we have that
and that
Taking , from relations (3.23)–(3.25) and (3.3) it follows that
Observing that , for all , from (3.26) and (3.4) we successively deduce that
Setting , from relation (3.27) we deduce that
Step 3.
Let be such that , for all and let be such that the pair satisfies ( ). We consider the functions , given by
Using analogous arguments with those used in the Step 2 we obtain that and the pair satisfies ( ). Moreover, using Lemma 2.7 we have that , for all . Then, according to our hypothesis we deduce that
Since , for all , using (3.30) and (3.5) we obtain that
Observing that
from (3.31) and (3.32) we deduce that
where .
Finally, from Steps 1–3 and relations (3.28) and (3.33) we deduce that the pair is uniformly admissible for the skewproduct flow . By applying Theorem 2.9 we conclude that is uniformly exponentially trichotomic.
The natural question arises whether the integral admissibility given by Definition 3.1 is also a necessary condition for the existence of the uniform exponential trichotomy. To answer this question, in what follows, our attention will focus on the converse implication of the result given by Theorem 3.3. Specifically, our study will motivate the admissibility concept introduced in this paper and will point out several qualitative aspects. First of all, we prove a technical result.
Proposition 3.4.
Let be a skewproduct flow which is uniformly exponentially trichotomic with respect to the families of projections . Let , let , and let be such that the pair satisfies ( ). We denote
and let denote the inverse of the operator
, for all . The following assertions hold:

(i)
the functions and have the following representations

(ii)
for every there is which does not depend on or such that
Proof.
Since there is such that . Let be given by Definition 2.4 and let .

(i)
Since the pair satisfies ( ) we have that
Since from (3.38) we have that , for all . Let . Then we deduce that
For in (3.39) we obtain that , for all . This shows that relation (3.35) holds for all . For from (3.38) we have that
so relation (3.35) holds for every .
For every from (3.38) we have that . This implies that
so we deduce that
From relation (3.42) it follows that , for all . In particular, we have that relation (3.36) holds for . For from (3.38) we obtain that
which implies that
for all . Thus, we conclude that relation (3.36) holds for every .

(ii)
Let and let . Setting
and using Hölder's inequality we deduce that
The second main result of the paper is as follows.
Theorem 3.5.
Let be a skewproduct flow. If is uniformly exponentially trichotomic, then the pair , is uniformly admissible for .
Proof.
Let be two constants and let , be the families of projections given by Definition 2.4. We set . For every and every we denote by the inverse of the operator .
For every and every we denote by
Then , for all and all and , for all .
Let and let be given by Proposition 3.4. We prove that all the properties from Definition 3.1 are fulfilled.
Let .
Step 1.
Let , . We consider the functions defined by
Since we have that and are correctly defined and continuous. Let be such that . Setting , and , we have that and , for .
We observe that , for all , which implies that
In addition, we have that , for all . This implies that
Since is continuous from relations (3.48) and (3.49) it follows that . Using similar arguments we deduce that . An easy computation shows that the pairs and satisfy ( ).
If , then we take .
Step 2.
Let and let be such that the pair satisfies ( ).
Suppose that . From Proposition 3.4 we have that
Let be such that . Since
for we deduce that . Since we obtain that
so . This implies that , for all . Moreover, using (3.51), for we have that
which implies that
Then, we obtain that
Setting from relations (3.50) and (3.55) it follows that
The case can be treated using similar arguments with those used above.
Step 3.
Let be such that , for all and let be such that the pair satisfies ( ).
Since and , for all , using Lemma 2.7 we deduce that , for all . Since the pair satisfies ( ) we obtain that
Let . Since , using relation (3.57) we have that
so , for all . This shows that, in this case, .
If is given by Step 2, we deduce that
and the proof is complete.
The central result of this paper is as follows.
Theorem 3.6.
A skewproduct flow is uniformly exponentially trichotomic if and only if the pair is uniformly admissible for .
Proof.
This follows from Theorems 3.3 and 3.5.
Remark 3.7.
The above result establishes for the first time in the literature a necessary and sufficient condition for the existence of the uniform exponential trichotomy of skewproduct flows, based on an inputoutput admissibility with respect to the associated integral equation. The chart described by our method allows a direct analysis of the asymptotic behavior of skewproduct flows, without assuming a priori the existence of a projection families, invariance properties or any reversibility properties. Moreover, the study is done in the most general case, without any additional assumptions concerning the flow or the cocycle.