König chain for compact matrix sets (Q5940034)

The author considers a compact set \(\Sigma\) of complex \((n\times n)\)-matrices. It is shown that \(\Sigma\) admits a so-called König chain, i.e., a sequence \(\{A_k \}\) in \(\Sigma\) such that \(\|A_1 A_2 \ldots A_k \|^{1/m}\) is greater or equal to the joint spectral radius of \(\Sigma\) for all operator norms \(\|\cdot\|\) [cf. \textit{G.-C. Rota} and \textit{W. G. Strang}, Nederl. Akad. Wet., Proc., Ser. A 63, 379-381 (1960; Zbl 0095.09701)].