An analogue to a result of Takahashi

In this paper we completely answer the following question: for given A∈B(H,K)A∈B(H,K), C∈B(L,K)C∈B(L,K) does there exist X∈B(H,L)X∈B(H,L) such that A+CXA+CX is injective? As an application of the obtained results, for given operators A∈B(H)A∈B(H), B∈B(K)B∈B(K) and C∈B(K,H)C∈B(K,H), separate necessary conditions and sufficient conditions on the triple (A,B,C)(A,B,C) for the existence of an operator X   such that the 2×22×2 operator matrix MX=[ACXB] is injective are presented.