The limit of the spectral radius of block Toeplitz matrices with nonnegative entries (Q1974668)

A bi-infinite sequence \( t= \{t_i\}_{i \in Z}\) of \(p \times p\) nonnegative valued matrices is given. It defines a sequence of block Toeplitz matrices \(T_n = (t_{ik})\), \(n = 1,2,\dots{}\), where \(t_{ik} = t_{k-i}\), \(i, k = 1,\dots{},n\). It is proved for \(T_n\) satisfying certain irreducibility conditions that the limit \(\mu(t) = \lim_{n_\rightarrow \infty} \mu_n(t)\), where \(\mu_n(t)\) is equal to the spectral radius \(\rho(T_n)\), is given by \(\inf \sigma(\xi)\), where \(\sigma(\xi) = \rho(\sum_{i \in Z} t_i \xi^i)\) and \(\xi\in [0, \infty]\).