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

TL;DR

This paper classifies weakly compact multiplication operators on L(L_p) for 1<p<, introduces -strictly singular operators, and explores their structure, revealing new embedding properties and answering longstanding questions.

Contribution

It provides a complete classification of weakly compact multiplication operators on L(L_p) and characterizes -strictly singular operators, addressing open problems from 1992 and 1976.

Findings

Operators on L_p with -strictly singularity have specific embedding properties.

On L_p for 1<p<2, -strictly singular operators restrict subspaces into L_r for all r<2.

For p=1, the results answer a question by Rosenthal from 1976.

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.