Compactification of a set of matrices with convergent infinite products
The author generalizes and unifies some known results on a set \(\Sigma\) of square matrices (real or complex) with right-convergent-products (RCPs). Here \(\Sigma\) is said to be an RCP set if for any sequence \((A_1, A_2,\dots)\in \Sigma^\infty\), \(\lim_{n\to\infty} A_1A_2\cdots A_n\) exists. A König chain of \(\Sigma\) with respect to a given norm \(\|\cdot\|\) is a sequence \((A_1,A_2,\dots)\in \Sigma^\infty\) such that \(\|A_1A_2\cdots A_m\|^{{1\over m}}\geq \widehat\rho\), \(m= 1,2,\dots\)\ . In Section 2 the author studies some important properties that are shared by both a bounded matrix set and its closure. In Section 3 he proves the existence of a König chain in a compact set of matrices, and then shows applications by reproducing some major known results.
scientific article; zbMATH DE number 1462183