On properties of the operator equation TT*=T+T*

In this paper, we study properties of the operator equation TT*=T+T* which T.T. West observed in [12]. We first investigate the structure of solutions T 2 B(H) of such equation. Moreover, we prove that if T is a polynomial root of solutions of that operator equation, then the spectral mapping theorem holds for Weyl and essential approximate point spectra of T and f(T) satisfies a-Weyl?s theorem for f?H(?(T)), where H(?(T)) is the space of functions analytic in an open neighborhood of ?(T).