Different completions of $A+CX$

Abstract

In this paper we use Takahasi’s idea of using properties of spectral measures in addressing the question of the existence of an operator $X$ such that $A+CX$ is of appropriate types. In particular, we consider the class of right semi-Fredholm operators. Also, in the case of right invertibility we will show how the results of this type can be used to address the appropriate completion problem of the operator matrix $M_X=\left[\begin{array}{cc}A& C\\ X& B\end{array}\right]$.