Let A, A_{1},  A_{2}, B, B_{1}, B_{2}, C_{1} and C_{2} be linear bounded operators on Hilbert spaces. In this paper, by using generalized inverses, we establish necessary and sufficient conditions for the existence of a common solution and give the form of the general common solution of the operator equations A_{1}XB_{1}=C_{1} and A_{2}XB_{2}=C_{2}, we apply this result to determine new necessary and sufficient conditions for the existence of Hermitian solutions  and give the form of the general Hermitian solution to the operator equation AXB=C. As a consequence, we give necessary and sufficient condition for the existence of Hermitian solution to the operator equation AXA^{*}+BYB^{*}=C.

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 A1,A2∈B(F,E), B1,B2∈B(D,G)B1,B2∈B(D,G). Let MA1,B1:X→A1XB1 be the multiplication operator on B(G,F) induced by A1,B1. In particular, LA1=MA1,I and RB1=MI,B1, where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B(G,F)B(G,F) is the sum of two multiplication operators Ψ=MA1,B1+MA2,B2. This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations MA1,B1(X)=C1 and MA2,B2(X)=C2 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 MA,B(X)=C. As a consequence, we obtain well-known results of Dajic´ and Koliha.