Splitting of Operators and Operator Equations

In this paper, we consider the splitting operators into the product of two positive operators. For given operators A and T E t3(7-1) with A = 0 and R(A) = R(T), if T can be splitted as T = AX for some positive operator X, then the existence of X and the properties of T are studied. The related research are the solutions of operator equations T = XAX and TX = XAX, which have some particular properties and broad applications in many fields. The conditions for the existence of solutions or idempotent solutions of these kinds of equations are studied and new representations of the general solutions are given.