In this work, the notion of extended eigenvalues a 2 ? lower triangular operator matrix has been researched. More precisely, relations between spectrum with spectrum, point and its diagonal entries have investigated. The obtained results supplemented by examples. addition, some properties block matrices displayed.

Denote by $$T_n^d(A)$$ an upper triangular operator matrix of dimension n whose diagonal entries are given and the others unknown. In this article, we provide necessary sufficient conditions for various types Fredholm Weyl completions . As consequences, recover many known results Zhang Wu (J Math Anal Appl 392(2):103–110, 2012) case $$n=2$$ already existing in literature, as well some perturbat...