Strictly Singular and Cosingular Multiplications

Lindström, Mikael ; Saksman, Eero ; Tylli, Hans-Olav

Canadian journal of mathematics, Tome 57 (2005) no. 6, pp. 1249-1278

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-050-7

Mots-clés : 47B47, 46B28

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