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

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.