### On Operator Equation TA-AT*=C

#### Abstract

Let H be a separable complex Hilbert space, and let B(H) be the algebra of bounded linear operators on H. Let *A, B, C* be elements in B(H). The equation *AX*-*XB*=*C*, which is called Sylvester’s equation or Rosenbium’s equation has been studied extensively by many mathematicians (see for example Bhatia and Rosenthal, 1997).

More recently, equations of the form *TA*-*AT*^{*}=*C*, where *T ^{*}* represents the adjoint of

*T*, began to receive more and more attention (Molnár, 1996; Molnár, 1994; Molnár, 1996; Semrl, 1994; Semrl, 1999; Semrl, 1991).

In this paper, we give necessary conditions for equation of the form *TA*-*AT*^{*}=ƒ(A) to have a solution in case A is a normal operator.

#### Keywords

Normal Operator, The Equation TA-AT*=C

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