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

Campbell, Stephen L.; Gellar, Ralph

doi:10.1090/S0002-9939-1976-0417841-3

Spectral properties of linear operators for which $T^T$ and $T$ $+$ $T^$ commute
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by Stephen L. Campbell and Ralph Gellar PDF

Proc. Amer. Math. Soc. 60 (1976), 197-202 Request permission

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.

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