Application of Abstract Semigroup Theory to the Asymptotic Behavior of $C_0$-Semigroups

Abstract

This study provides a general approach, how solutions of a Cauchy problem leading to bounded $C_0-$semigroups can be splitted into a sum of a reversible, sometimes almost periodic and a signal noise part, in cases when the relative weak compactness of the orbit needed for the application of the deLeeuw–Glicksberg theory is not given. Especially, a splitting for the Ornstein–Uhlenbeck-semigroup for a class of continuous and bounded functions is given. It is dicussed, when splittings becomes unique, how close almost periodicity is, and how spectral theory comes into play. Further, in some general cases the question, when the reversible part becomes almost periodic is answered.

Appendix A

1.1 On Idempotents

In this section, we recall the main results on right semitopological semigroups, which are used to obtain the splittings and the group (sub)structures. As previously mentioned, in this section S is only right(left)-semitopological and not necessarily Abelian.

Definition A.1

([44, p. 12]). The set of idempotents in a right(left)-semitopological semigroup S is denoted by E(S). On E(S) we define the relations $\le _L$ and $\le _R$ by

$$\begin{aligned} e\le _L f&\text{ if }&ef=e, \\ e\le _R f&\text{ if }&fe=e. \end{aligned}$$

If e and f commute, then we omit the indices L and R.

Definition A.2

Let $(A,\le )$ be a set with a transitive relation. Then, an element a is called $\le -$maximal [−minimal] in A if for every $a^{\prime }\in A$, $a\le a^{\prime }$ implies $a^{\prime } \le a$ [$a^{\prime } \le a$ implies $a\le a^{\prime }$].

1.2 Topological Vector Spaces

Next, we recall some consequences of [29, Corollary 2 (a), p. 127]. In particular, we have the following.

Proposition A.3

Let $E\subset F\subset X^*$ and $\tau$ be a locally convex topology on $X^*$, where $\sigma (X^*,X)\subset \tau \subset { \displaystyle \left\| {\cdot } \right\| } .$ If no vector of E can be separated from F by a $\tau -$continuous functional, then $\overline{E}^{\tau }=\overline{F}^{\tau }.$

Proof

If $\overline{E}^{\tau }\not =\overline{F}^{\tau },$ then there exists an $x\in \overline{F}^{\tau } \backslash \overline{E}^{\tau }$ and a $\tau -$continuous functional $\phi$ such that $\phi _{|\overline{E}^{\tau }}=0$ and $\phi (x)=1.$ By definition, we have for net ${ \left\{ { {x}_{\lambda } } \right\} _{{\lambda } \in {\Lambda }} }\subset F$ the $\tau$-convergence $x_{\lambda }\rightarrow x.$ Moreover, we find a subnet that has no intersection with $\overline{E}^{\tau }.$ The continuity $\phi$ leads to an element $x_{\lambda _0}$ with $\phi (x_{\lambda _0})>1/2,$ which illustrates the contradiction. $\square$