s-numbers of elementary operators

Ivan Todorov (Queen's University, Belfast)

Tuesday 28th October, 2008 15:00-16:00 214


Let $B(H)$ be the algebra of all bounded linear operators acting on a Hilbert space $H$. If $A,B\in B(H)$ let $M_{A,B}$ be the operator on $B(H)$ given by $M_{A,B}(X) = AXB$, $X\in B(H)$. An {\it elementary operator} $\Phi$ on $B(H)$ is an operator of the form $\Phi = \sum_{i=1}^m M_{A_i,B_i}$, where $A_i,B_i$, $i = 1,\dots,m$ are bounded linear operators (called {\it symbols} of $\Phi$). A well-known theorem of Fong and Sourour states that if $\Phi$ is an elementary operator on $B(H)$ then $\Phi$ is compact if and only if its symbols $A_i,B_i$, $i = 1,\dots,m$, can be chosen to be compact. In this talk we will give quantitative versions of this result using the notions of an $s$-number function introduced by A. Pietsch and the theory of ideals of $B(H)$ developed by von Neumann, R. Schatten and W. Calkin. We will relate the behaviour of the $s$-numbers of a given elementary operator to that of its symbols. We will further extend these results to the case of elementary operators acting on general C*-algebras. The talk is based on a joint work with M. Anoussis and V. Felouzis.

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