Weak operator topology, operator ranges and operator equations via Kolmogorov widths

Details Weak operator topology, operator ranges and operator equations via Kolmogorov widths Weak operator topology, operator ranges and operator equations via Kolmogorov widths

2009-02-19

arxiv.org/abs/0902.3483v1

Integral Equations and Operator Theory 65 (2009), 551-572

Mathematics

Functional Analysis

Scientific paper

10.1007/s00020-009-1691-0

Let $K$ be an absolutely convex infinite-dimensional compact in a Banach space $\mathcal{X}$. The set of all bounded linear operators $T$ on $\mathcal{X}$ satisfying $TK\supset K$ is denoted by $G(K)$. Our starting point is the study of the closure $WG(K)$ of $G(K)$ in the weak operator topology. We prove that $WG(K)$ contains the algebra of all operators leaving $\overline{\lin(K)}$ invariant. More precise results are obtained in terms of the Kolmogorov $n$-widths of the compact $K$. The obtained results are used in the study of operator ranges and operator equations.

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