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Strictly Singular and Cosingular Multiplications

Published online by Cambridge University Press:  20 November 2018

Mikael Lindström ,
Eero Saksman  and
Hans-Olav Tylli
Mikael Lindström
Affiliation:
Department of Mathematics, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500 Åbo, Finland, email: mlindstr@abo.fi
Eero Saksman
Affiliation:
Department of Mathematics and Statistics, P.O. Box 35 (MaD), FIN-40014 University of Jyväskylä, Finland, email: saksman@maths.jyu.fi
Hans-Olav Tylli
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, Gustaf Hällströmin katu 2b, FIN-00014 University of Helsinki, Finland, email: hojtylli@cc.helsinki.fi
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Abstract

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Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity and cosingularity of the two-sided multiplication operators $S\,\mapsto \,ASB$ on $L(X)$, where $A,\,B\,\in \,L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1\,<\,p\,<\,\infty $ and $p\,\ne \,2$. Our main result establishes that the multiplication $S\,\mapsto \,ASB$ is strictly singular on $L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ if and only if the non-zero operators $A,\,B\,\in \,L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ are strictly singular. We also discuss the case where $X$ is a ${\mathcal{L}^{1}}-$ or a ${{\mathcal{L}}^{\infty }}-$space, as well as several other relevant examples.

Keywords

47B4746B28
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 6 , 01 December 2005 , pp. 1249 - 1278
Copyright
Copyright © Canadian Mathematical Society 2005

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