Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute. II

Campbell, Stephen L.; Gellar, Ralph

doi:10.1090/S0002-9947-1977-0435905-0

Linear operators for which $T^T$ and $T+T^$ commute. II
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by Stephen L. Campbell and Ralph Gellar PDF

Trans. Amer. Math. Soc. 226 (1977), 305-319 Request permission

Abstract:

Let $\theta$ denote the set of bounded linear operators T, acting on a separable Hilbert space $\mathcal {K}$ such that ${T^\ast }T$ and $T + {T^\ast }$ commute. It is shown that such operators are ${G_1}$. A complete structure theory is developed for the case when $\sigma (T)$ does not intersect the real axis. Using this structure theory, several nonhyponormal operators in $\theta$ with special properties are constructed.

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