ON THE BANACH-STONE PROBLEM FOR $L^p$-SPACES

Abstract

The Banach-Stone problem for $L^p$-spaces is to assert when a linear isometry between $L^p$-spaces is a weighted composition operator. We shall show that every $\sigma$-finite measure space with Sikorski’s property solves the Banach-Stone probelm. In addition, we show that if $X$ is a totally ordered and Dedekind complete, then every $\sigma$-finite $\mu$-separable measure space $(X,\mathcal{B},\mu)$ has Sikorski’s property.