Weak operator topology, operator ranges and operator equations via Kolmogorov widths (Q2655382)

Let \(K\) be an absolutely convex infinite-dimensional compact in a Banach space \(X\). The set of all bounded linear operators \(T\in L(X)\) satisfying \(TK \supseteq K\) is denoted by \(G(K)\), which is closed in the norm topology and sequentially closed in the weak operator topology. In general, the weak operator closure \(WG(K)\) of \(G(K)\) is much larger than \(G(K)\) and, in many cases, it coincides with the algebra \(L(X)\). The authors address the question ``How large is \(G(K)\)?'' They describe \(WG(K)\) and the ultra-weak closure of \(G(K)\) (for the case of Hilbert spaces). Indeed, they show that \(WG(K)\) contains the algebra of all operators leaving the closed linear span of \(K\) invariant. They also deal with the set \(G(K_1, K_2)\) of all operators \(T\) which have the property \(TK_1\supseteq K_2\), where \(K_1, K_2\) are fixed convex compacts in some Hilbert spaces. Some applications to the theory of quadratic operator inequalities and operator fractional linear relations are presented as well.