Abstract:

Using the block operator matrix technique and the generalized inverse of operator,the solutions to the operator equation AXB=C are studied in infinite Hilbert space. The sufficient and necessary condition for the existence of solutions to the equation AXB=C and the representation of the solutions are established. Especially when R(A*)R(B) or R(B)R(A*), the sufficient and necessary condition for the existence of positive solutions of the equation AXB=C and the general form of the positive solutions are also derived.

Key words: operator equation; positive operator; Moore-Penrose inverse