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 ∈ 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(σ(T)), where H(σ(T)) is the space of functions analytic in an open neighborhood of σ(T).