Perturbing the spectrum of operator $T_n^d(A)$

TL;DR

This paper characterizes the invertibility of a class of upper triangular operator matrices using only their diagonal entries, generalizing previous results and removing the separability assumption.

Contribution

It provides new characterizations of invertibility for operator matrices without assuming separability, extending and correcting prior work, and offers n-dimensional analogues.

Findings

Characterizations of invertibility in terms of diagonal entries.

Generalization of previous results to non-separable spaces.

Recovery of perturbation results in Hilbert space setting.

Abstract

Let $T_n^d(A)$ denote a partial upper triangular operator matrix whose diagonal entries are given and the others unknown. In this article we have aim to find characterizations of (left,right) invertibility of $T_n^d(A)$ in terms of diagonal entries solely, and hence we provide statements which generalize and correct results of Zhang S., Wu Z. (2012). We pose our results without invoking separability condition, thus improving results of Zhang S., Wu Z. (2012), and we give appropriate n-dimensional analogues without assuming separability as well. We recover many perturbation results of Djordjevi\'c D. S. (2002), and obtain some results of Du H. K., Pan J. (1994) and Han J. K., Lee H. Y., Lee W. Y. (2000) in the case of the Hilbert space setting.