Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)+ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈LN ×  N∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)+|| > ||a−1||∞ holds. Moreover, we discuss the asymptotic behavior of ||T(tra)+|| for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\, \in \,\user2{\mathbb{Z}}$$\end{document}. The results are illustrated by numerical experiments.