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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spectral approximations of a normal operator


Author: Richard Bouldin
Journal: Proc. Amer. Math. Soc. 76 (1979), 279-284
MSC: Primary 47B15
MathSciNet review: 537088
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Abstract: If $ \Lambda $ is a closed convex set in the complex plane then $ \mathfrak{N}(\Lambda ;H)$ denotes all the normal (bounded linear) operators on the fixed separable Hilbert space H with spectrum contained in $ \Lambda $. The fixed operator A has N as an $ \mathfrak{N}(\Lambda ;H)$-approximant provided N belongs to $ \mathfrak{N}(\Lambda ;H)$ and the operator norm $ \left\Vert {A - N} \right\Vert$ equals $ {\rho _\Lambda }(A)$, the distance from A to $ \mathfrak{N}(\Lambda ;H)$. With some hypothesis on $ \Lambda $, this note proves that the dimension of the convex set of all $ \mathfrak{N}(\Lambda ;H)$-approximants of normal operator A is $ {(\dim {H_0})^2}$ where $ {H_0}$ is the orthogonal complement of $ \ker (\vert A - F(A)\vert - {\rho _\Lambda }(A))$ and $ F(z)$ is the unique distaince minimizing retract of the complex plane onto $ \Lambda $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0537088-2
PII: S 0002-9939(1979)0537088-2
Article copyright: © Copyright 1979 American Mathematical Society