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We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept of

The possibility of reducing the nonautonomous case in the study of associated evolution operators to the autonomous case of evolution semigroups on various Banach function spaces can be considered as an important way towards applications issued from the real world. Of great importance in the study the solution of differential equations is the approach by evolution families, as the techniques from the domain of non autonomous equations with unbounded coefficients in infinite dimensions can be extended in this direction.

Appropriate for the study of evolution equations in infinite dimensions are also the skew-evolution semiflows, introduced by us in [

The techniques used in the investigation of exponential stability and exponential instability were generalized for the case of exponential dichotomy in [

Let

A mapping

A mapping

The mapping

Let

If

We consider that

A particular class of skew-evolution semiflows is emphasized in the following.

Let us consider a skew-evolution semiflow

We observe that

Let

Let

We denote that

which generates a

The mapping

is a classic cocycle over the semiflow

is a skew-evolution semiflow on

More directly, if

then

Let us recall the definition of a semigroup of linear operators, and let us give an example which shows that this is generating a skew-evolution semiflow.

A mapping

One can naturally associate to every semigroup of operators the mapping

Other examples of skew-evolution semiflows are given in [

A skew-evolution semiflow

We intend to give a new approach for the property of trichotomy for skew-evolution semiflows, the

Let

is called

A projections family

The splitting of the state space into three subspaces will be assured by the following.

Three projections families

In what follows we introduce the elements which will allow us to introduce a new concept of trichotomy for skew-evolution semiflows. We consider a mapping

For every function

Let us denote that

A skew-evolution semiflow

such that

for all

Let

hold for all

satisfy relations (

For

In what follows, if

for every

A skew-evolution semiflow

for all

such that relations (i)–(iii) of Definition

As

Hence, relation

We obtain the relations

Hence, relations (i)–(iii) of Definition

Proposition

We obtain a characterization for the property of trichotomy, by means of the shifted skew-evolution semiflow.

A skew-evolution semiflow

the evolution cocycle

the evolution cocycle

there exists a constant

Let us consider that

for all

If we consider that

for all

Further, we obtain

for all

the first relation in

If there exists a constant

for all

Another characterization for the property of

A

there exists

there exists

there exist

An equivalent relation is obtained, if we consider Definition

for all

for all

(ii) According to Definition

The property of function

for all

(iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Theorem

Let

For every

Hence, we have that

For

which implies that

Further, the following relations hold:

By integrating on

and further

which implies that

(ii) Let

According to the fact that

As the evolution cocycle

and further

For

and, if

for all

with the property

(iii) A similar proof for the

If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with

We define

The property described by relation (

This paper was conceived during the visit at the Institute of Mathematics of Bordeaux, France. The author wishes to express her profound and respectful gratitude to Professor Bernard Chevreau and Professor Mihail Megan. Also, the author gratefully acknowledges helpful comments and suggestions from the referees.