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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The similarity orbit of a normal operator


Author: L. A. Fialkow
Journal: Trans. Amer. Math. Soc. 210 (1975), 129-137
MSC: Primary 47A55; Secondary 47B15
DOI: https://doi.org/10.1090/S0002-9947-1975-0374956-X
MathSciNet review: 0374956
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Abstract: If $ N$ is a bounded normal operator on a separable Hilbert space $ \mathcal{H}$, let $ \mathcal{S}(N)$ denote the similarity orbit of $ N$ in $ L(\mathcal{H})$ and let $ {\mathcal{S}_k}(N)$ denote the set of all compact perturbations of elements of $ \mathcal{S}(N)$. It is proved that $ \mathcal{S}(N)({\mathcal{S}_K}(N))$ is norm closed in $ L(\mathcal{H})$ if and only if the spectrum (essential spectrum) of $ N$ is finite. If the essential spectrum of $ N$ is infinite and $ M$ is a normal operator whose spectrum is connected and contains that of $ N$, then $ M$ is in the closure of $ \mathcal{S}(N)$. If the spectrum of $ N$ is connected, this result characterizes the normal elements of the closure of $ \mathcal{S}(N)$. A normal operator is similar to a nonquasidiagonal operator if and only if its essential spectrum contains more than two points.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0374956-X
Article copyright: © Copyright 1975 American Mathematical Society