This is an announcement for the paper "Weak operator topology, operator ranges and operator equations via Kolmogorov widths" by M.I. Ostrovskii and V.S. Shulman. 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. Archive classification: math.FA math.OA Mathematics Subject Classification: 47A05; 41A46; 47A30; 47A62 The source file(s), ostshu.tex: 68035 bytes, is(are) stored in gzipped form as 0902.3483.gz with size 21kb. The corresponding postcript file has gzipped size 139kb. Submitted from: ostrovsm@stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.3483 or http://arXiv.org/abs/0902.3483 or by email in unzipped form by transmitting an empty message with subject line uget 0902.3483 or in gzipped form by using subject line get 0902.3483 to: math@arXiv.org.