The essential norm of multiplication operators on $L_p(\mu)$

TL;DR

This paper extends the known formula for the essential norm of multiplication operators on $L_p(u)$ spaces to include the case when p=1, providing a unified proof for all p in [1, ).

Contribution

It proves that the essential norm formula for multiplication operators on $L_p$ spaces holds for p=1 and offers a unified proof for all p in [1, ).

Findings

The essential norm formula applies to $L_1$ spaces.

A unified proof for the formula valid for all p in [1, ).

Extension of known results to the case p=1.

Abstract

We show that the formula for the essential norm of a multiplication operator on $L_p$, for $1<p<\infty$, also holds for $p=1$. We also provide a proof for the formula which works simultaneously for all $p\in[1,\infty)$.