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arXiv · 0708.0560

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

William B. Johnson·Gideon Schechtman
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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.

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William B. Johnson, Gideon Schechtman. 2007-08-03. Multiplication operators on L(L_p) and $\ell_p$-strictly singular operators. https://arxiv.org/abs/0708.0560

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