On products of unbounded collections of matrices (Q1906769)

The authors discuss problems on the set \({\mathcal B}\) of all complex \(n \times n\) matrices with bounded products, and prove that if \({\mathcal B}^k : = \{B_1 B_2 \dots B_k : B_i \in {\mathcal B}\}\) is bounded for some \(k\), then \({\mathcal B}^m\) is bounded for all \(m \geq n\). Using this result they extend the relation \(\limsup_{k \to \infty} \widehat \rho_k ({\mathcal B})^{1/k} \leq \widehat \rho_k ({\mathcal B})^{1/k}\) to the unbounded family, where \(\widehat \rho_k ({\mathcal B}) = \sup \{|B_1 B_2 \dots, B_k |: B_i \in {\mathcal B} (i = 1, 2, \dots, k)\}\).