Invertibility of linear combinations in B(ℋ)

Abstract

In this paper we use the results concerning the invertibility of an upper triangular operator matrix in order to obtain the necessary and sufficient conditions for the invertibility of the linear combination , for given operators and . We present necessary and sufficient conditions for the invertibility of in some special cases such as that when A, B are projectors, orthogonal projectors, when , when and when either one of A or B is injective. Also, we consider the cases when the invertibility of is independent of the choice of the scalars .