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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Topological properties of paranormal operators on Hilbert space


Author: Glenn R. Luecke
Journal: Trans. Amer. Math. Soc. 172 (1972), 35-43
MSC: Primary 47B20; Secondary 47B99
MathSciNet review: 0308839
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Abstract: Let $ B(H)$ be the set of all bounded endomorphisms (operators) on the complex Hilbert space $ H.T \in B(H)$ is paranormal if $ \vert\vert{(T - zI)^{ - 1}}\vert\vert = 1/d(z,\sigma (T))$ for all $ z \notin \sigma (T)$ where $ d(z,\sigma (T))$ is the distance from $ z$ to $ \sigma (T)$, the spectrum of $ T$. If $ \mathcal{P}$ is the set of all paranormal operators on $ H$, then $ \mathcal{P}$ contains the normal operators, $ \mathfrak{N}$, and the hyponormal operators; and $ \mathcal{P}$ is contained in $ \mathcal{L}$, the set of all $ T \in B(H)$ such that the convex hull of $ \sigma (T)$ equals the closure of the numerical range of $ T$. Thus, $ \mathfrak{N} \subseteq \mathcal{P} \subseteq \mathcal{L} \subseteq B(H)$. Give $ B(H)$ the norm topology. The main results in this paper are (1) $ \mathfrak{N},\mathcal{P}$, and $ \mathcal{L}$ are nowhere dense subsets of $ B(H)$ when $ \dim H \geq 2$, (2) $ \mathfrak{N},\mathcal{P}$, and $ \mathcal{L}$ are arcwise connected and closed, and (3) $ \mathfrak{N}$ is a nowhere dense subset of $ \mathcal{P}$ when $ \dim H = \infty $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0308839-5
PII: S 0002-9947(1972)0308839-5
Keywords: Hilbert space, resolvent, normal operator, convexoid operators, properties and topological properties of $ {G_1}$, operators
Article copyright: © Copyright 1972 American Mathematical Society