This is an announcement for the paper "Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators" by William B. Johnson and Gideon Schechtman.

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.

Archive classification: math.FA

Mathematics Subject Classification: 46B20; 46E30

The source file(s), JSElemOpAug3.07.tex: 53364 bytes, is(are) stored in gzipped form as 0708.0560.gz with size 17kb. The corresponding postcript file has gzipped size 120kb.

Submitted from: gideon@weizmann.ac.il

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