An analogue to a result of Takahashi II

In this paper we address the question of the existence of an operator X such that A+CX is of various appropriate types: a closed range, right invertible, left invertible or a Fredholm operator. We point out to a connection between the existence of such X and the problem of completion of operator matrices of type [GRAPHICS] is an element of B(H circle plus H) and present necessary and sufficient conditions for the operator matrix [GRAPHICS] is an element of B(H circle plus H) to be completed to a Fredholm operator. Also, under certain conditions we solve the corresponding completion problem to a closed range operator. (C) 2019 Published by Elsevier Inc.