JournalsjemsVol. 10 , No. 4DOI 10.4171/jems/141

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

Abstract

A classification of weakly compact multiplication operators on L(Lp)L(L_p), 1<p<1<p<\infty, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of p\ell_p-strictly singular operators, and we also investigate the structure of general p\ell_p-strictly singular operators on LpL_p. The main result is that if an operator TT on LpL_p, 1<p<21<p<2, is p\ell_p-strictly singular and TXT_{|X} is an isomorphism for some subspace XX of LpL_p, then XX embeds into LrL_r for all r<2r<2, but XX need not be isomorphic to a Hilbert space. It is also shown that if TT is convolution by a biased coin on LpL_p of the Cantor group, 1p<21\le p <2, and TXT_{|X} is an isomorphism for some reflexive subspace XX of LpL_p, then XX is isomorphic to a Hilbert space. The case p=1p=1 answers a question asked by Rosenthal in 1976.