Bands in $L_p$-spaces

TL;DR

This paper characterizes bands in $L_p$-spaces over general measure spaces by decomposing the measure into parts corresponding to each band, with applications to absorption semigroups.

Contribution

It provides a measure-theoretic decomposition that characterizes all bands in $L_p$-spaces, extending the understanding of their structure.

Findings

Every band in $L_p(\mu)$ corresponds to a measure decomposition.

The theory is illustrated with a concrete example.

Application to absorption semigroups demonstrates practical relevance.

Abstract

For a general measure space $(\Omega,\mu)$, it is shown that for every band $M$ in $L_p(\mu)$ there exists a decomposition $\mu=\mu'+\mu^{\prime\prime}$ such that $M=L_p(\mu')=\{f\in L_p(\mu);f=0\ \mu^{\prime\prime}\text{-a.e.}\}$. The theory is illustrated by an example, with an application to absorption semigroups.