Transient behavior of powers and exponentials of large Toeplitz matrices
The paper summarizes several known results, giving a good introduction to the field of powers and exponential of large Toeplitz matrices. The notion of polynomial numerical hull is reviewed with the aim of getting estimates of the norms. The role of pseudospectra in connection with the norms of powers of matrices is developed and some known results on the asymptotic behavior of the norms of Toeplitz-like matrices are collected. Examples show the application of the different results given in the paper. Moreover, the paper has new interesting results. In contrast with asymptotic results, different genuinely finite results are presented. For the Laurent polynomials it is proved that the lower right boundary of the sky region is always sublinear and in many cases superlinear. An analogous weaker result for exponentials of large Toeplitz matrices is established, having in mind that \(\exp{( \tau T_{n}(a))}\) has critical behavior if and only if \(\sup \text{Re}(a) >0\). The main challenge of the paper is to show that powers of large Toeplitz matrices have critical behavior: the norms of the powers grow exponentially to infinity or run through a critical transient phase before decaying exponentially to zero if and only if the \(L_\infty\) norm of the symbol is greater than one.
scientific article; zbMATH DE number 2156114