On the Solvability of the Operator Equation AT+T*B=C

In this paper we study the solvability of the operator equation AT+T*B=C for linear bounded operators on Hilbert space, where T is the unknown operator. First we show this equation may not have a solution in B(H), and then we show that this equation have a simple solution in case A,B are Hermitian or Skew-Hermitian operators. Finaly we study the solvability of the operator equation AT+T*B=f(A) in case A and B are normal operators.