Multiplication operators on L^p

TL;DR

This paper studies multiplication operators on L^p spaces, showing they are compact perturbations of diagonal operators and classifying them up to similarity and approximate similarity.

Contribution

It provides a classification of multiplication operators on L^p spaces up to similarity and approximate similarity, highlighting their structure as compact perturbations of diagonal operators.

Findings

Operators are compact perturbations of diagonal operators.

Classification up to similarity modulo compact operators.

Classification up to approximate similarity.

Abstract

We show that every operator on $L^{p}$, $1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these operators up to similarity modulo compact operators and up to approximate similarity.