Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators

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.