# Multiplication operators on <em>L</em>(<em>L<sub>p</sub></em>) and <em>l<sub>p</sub></em>-strictly singular operators

### William B. Johnson

Texas A&M University, College Station, United States### Gideon Schechtman

Weizmann Institute of Science, Rehovot, Israel

## Abstract

A classification of weakly compact multiplication operators on $L(L_p)$, $1<p<\infty$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $\ell_p$-strictly singular operators, and we also investigate the structure of general $\ell_p$-strictly singular operators on $L_p$. The main result is that if an operator $T$ on $L_p$, $1<p<2$, is $\ell_p$-strictly singular and $T_{|X}$ is an isomorphism for some subspace $X$ of $L_p$, then $X$ embeds into $L_r$ for all $r<2$, but $X$ need not be isomorphic to a Hilbert space. It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p <2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.