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

Abstract

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 is an element of 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 2 H(sigma(T)), where H(sigma(T)) is the space of functions analytic in an open neighborhood of sigma(T).