Asymptotics of the Smallest Singular Value of a Class of Toeplitz-generated Matrices II

Square matrices of the form Xn=Tn+fn(Tn-1)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X_n = T_n + f_n(T_n^{-1})^*}$$\end{document}, where Tn is a n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \times n}$$\end{document} invertible banded Toeplitz matrix and fn some positive sequence are considered. Convergence via an order estimate is proven for the difference of ‖Xn-1‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\|X_n^{-1}\|}$$\end{document} and a function depending only on fn. Fredholmness of the infinite counterpart of Tn is shown to greatly affect this result. A correction of a proof in the paper on which the current research is based, is appended as well.