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

#### Abstract

In this paper, we study properties of the operator equation $TT^{\ast}=T+T^{\ast}$ shich T.T. West obseved in \cite{West}. We first investigate the structure of solutions $T\in B(\mathcal H)$ 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 $f\in H(\sigma(T))$, where $H(\sigma(T))$ is the space of functions analytic in an open neighborhood of $\sigma(T)$.

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