Invertibility properties of operator matrices on Hilbert spaces

Abstract

Denote by $$T_n^d(A)$$ an upper triangular operator matrix of dimension n whose diagonal entries are given and the others are unknown. In this article, we provide necessary and sufficient conditions for various types of Fredholm and Weyl completions of $$T_n^d(A)$$ . As consequences, we recover many known results of Zhang and Wu (J Math Anal Appl 392(2):103–110, 2012) for the case $$n=2$$ already existing in the literature, as well as some perturbation results of Wu and Huang (Acta Math Sin (Engl Ser) 36(7):783–796, 2020; Ann Funct Anal 11(3):780–798, 2020) given for the case of arbitrary $$n\ge 2$$ .