Invertible and regular completions of operator matrices

Abstract

In this paper, for given operators A ∈ B(X) and B ∈ B(Y), the set of all C ∈ B(Y,X) such that the operator matrix M_C = \left[ \begin{array}{cc} A & C \\ O & B \end{array} \right] is injective, invertible, left invertible and right invertible, is described. Answers to some open questions are given. Also, in the case when A and B are relatively regular operators, the set of all C ∈ B(Y,X) such that M_C is regular is described. In addition, a necessary and a sufficient conditions are given for MC to be regular with the inner inverse of a certain given form.