Spectral properties of linear operators for which $T\sp*T$ and $T$ $+$ $T\sp*$ commute

Abstract

The class of linear operators for which ${T^\ast }T$ and $T + {T^\ast }$ commute is studied. It is shown that such operators are normaloid. If T is also completely nonnormal, then $\sigma (T) = \sigma ({T^\ast })$. Also, isolated points of $\sigma (T)$ are reducing eigenvalues. Finally, if $\sigma (T)$ is a subset of either a vertical line or the real axis, then T is normal.

Locations

• Proceedings of the American Mathematical Society - View - PDF