On Compactifications of bounded $C_0-$semigroups

TL;DR

This paper refines the compactification of bounded $C_0$-semigroups, reducing the operator space involved and extending ergodic results to dual semigroups using adjoint theory.

Contribution

It introduces a more efficient compactification method for bounded $C_0$-semigroups and extends ergodic theorems to dual semigroups.

Findings

Reduced the operator space for compactification.

Decomposed $X^{igcirc}$ using the new approach.

Extended ergodic results to dual semigroups.

Abstract

In this study, we refine the compactification presented by Witz \cite{Witz} for general semigroups to the case of bounded $C_0$-semigroups, involving adjoint theory for this class of operators. This approach considerably reduces the operator space in which the compactification is performed. Additionally, this approach leads to a decomposition of $X^{\sun}$ and to an extension of ergodic results to dual semigroups.