Short notes on $L^1(\Omega,X)$ with infinite measure

TL;DR

This paper investigates duality, weak compactness, and stability properties of $L^1(\Omega,X)$ spaces with infinite measure, extending existing results and applying them to almost periodicity and evolution semigroups.

Contribution

It provides new duality results, criteria for weak compactness, and stability conditions for $L^1(\Omega,X)$ spaces with infinite measure, extending previous work and removing the approximation property requirement.

Findings

Dual of $L^1(\mu,X)$ characterized in positive and $\sigma$-finite cases

Necessary and sufficient criteria for weak compactness in $L^1(S,\mu,X)$

Weak compactness implies strong stability for evolution semigroups

Abstract

This study uses the ideas of \cite{Rieffel} to provide the dual of $L^1(\mu,X)$ in the positive and $\sigma-$ finite cases. This results in elegant necessary and sufficient criteria for weak compactness in $L^1(S,\mu,X)$ in the $\sigma-$finite case, using the ideas of \cite{RuessL1} and \cite{Cooper}. Finally, the result of \cite{NeervenLNM} is extended to compute the sun-dual of $L^1(\re,X)$ with respect to the canonical translation semigroup, dropping the approximation property from $X^*,$, which is applied to obtain almost periodicity for integrals of non-smooth functions. Moreover, for evolution semigroups, it is shown that weak compactness of the orbits implies strong stability.