Riesz projections for a non-hyponormal operator

Abstract

J. G. Stampfli proved that if a bounded linear operator $T$ on a Hilbert space $\mathscr H$ satisfies ($G_1$) property, then the Riesz projection $P_{\lambda}$ associated with $\lambda\in{\rm iso}\sigma(T)$ is self-adjoint and $P_{\lambda}\mathscr{H}=(T - \lambda)^{-1}(0)=(T^{*} - \bar{\lambda})^{-1}(0)$. In this note we show that Stampfli's result is generalized to an nilpotent extension of an operator having ($G_1$) property.