Operator equations and inner inverses of elementary operators.

Let E,F,G,D be infinite complex Banach spaces and B (F , E) the Banach space of all bounded linear operators from F into E. Consider A 1 , A 2 ∈ B (F , E) , B 1 , B 2 ∈ B (D , G). Let M A 1 , B 1 : X → A 1 X B 1 be the multiplication operator on B (G , F) induced by A 1 , B 1 . In particular, L A 1 = M A 1 , I and R B 1 = M I , B 1 , where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B (G , F) is the sum of two multiplication operators Ψ = M A 1 , B 1 + M A 2 , B 2 . This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations M A 1 , B 1 (X) = C 1 and M A 2 , B 2 (X) = C 2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation M A , B (X) = C. As a consequence, we obtain well-known results of Daji c ´ and Koliha. [ABSTRACT FROM AUTHOR]