Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices

Abstract

Given a continuous function a on the complex unit circle, let T(a) denote the infinite Toeplitz matrix generated by a and let Tn(a) stand for the (n+1)×(n+1) principal section of T(a). We think of T(a) and Tn(a) as operators on l2 spaces. A classical result by Gohberg and Feldman says that if T(a) is invertible, then so is Tn(a) for all sufficiently large n≥n0 and \(\). Only in 1994 did we realize that in fact \(\). In this paper, we provide estimates for the speed with which \(\) converges to \(\). We prove that in the “generic case” the convergence speed can be estimated by the smoothness of a, whereas in some “exceptional cases” (e.g., if T(a) is Hermitian or triangular) it is not the smoothness of $a$ but the orders of certain zeros which determine the convergence speed. Some of the results are extended to operators on lp spaces.