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

TL;DR

This paper investigates the weak operator topology closure of operators that contain a given compact set in their range, using Kolmogorov widths to analyze operator ranges and equations in Banach spaces.

Contribution

It characterizes the weak operator topology closure of certain operator sets and relates it to invariance properties and Kolmogorov widths, advancing understanding of operator ranges.

Findings

Closure of G(K) contains all operators leaving the span of K invariant

Results connect Kolmogorov widths with operator range properties

Applications to operator equations in Banach spaces

Abstract

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.