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

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 and Wu (J. Math. Anal. Appl. 392:103–110, 2012). We pose our results without invoking separability condition, thus improving results of Zhang and Wu (J. Math. Anal. Appl. 392:103–110, 2012), and we give appropriate n-dimensional analogues, without assuming separability as well. We recover many perturbation results of Djordjević (J. Oper. Theory 48:467–486, 2002), and obtain some results of Du and Pan (Proc. Amer. Math. Soc. 121: 761–766, 1994) and Han et al. (Proc. Amer. Math. Soc. 128:119–23, 2000) in the case of the Hilbert space setting.