Invertibility of linear combinations in B(ℋ)

On nonsingularity and group invertibility of combinations of two group invertible matrices

“…Recently, invertibility of the sum (the difference) of two given operators was considered in [7] with a special emphasis on some cases such as when A and B are idempotents, when R(A) ∩ R(B) = {0}, when A and B are injective, etc. In this section the main results concern the nonsingularity of the difference of two group invertible matrices as well as the nonsingularity of some forms under some assumptions.…”

Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

“…The motivation behind this paper are the papers [15] and [11] where the invertibility of the linear combination αA + βB was considered in the case when A, B ∈ B(H) are regular operators and α, β ∈ C \ {0}, and also some recently published papers (see [17,12,18,19,22,20]) which considered the independence of the invertibility of the linear combination αA + βB in the cases when A, B ∈ B(H) are projectors or orthogonal projectors (see [12,17,18,19,20,22] for results concerning specifically projections and orthogonal projections). The aim of this paper is to investigate injectivity of the linear combination αA + βB where A, B ∈ B(H) are given operators and α, β ∈ C \ {0}.…”