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

On the spectral picture of an irreducible subnormal operator


Author: Paul McGuire
Journal: Proc. Amer. Math. Soc. 104 (1988), 801-808
MSC: Primary 47B20; Secondary 47A10
MathSciNet review: 964860
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Abstract: This paper extends the following result of R. F. Olin and J. E. Thomson: A compact subset $ K$ of the plane is the spectrum of an irreducible subnormal operator if and only if $ \mathcal{R}(K)$ has exactly one nontrivial Gleason part $ G$ such that $ K$ is the closure of $ G$. The main result of this paper is that the only additional requirement needed for the pair $ \left\{ {K,{K_e}} \right\}$ to be the spectrum and essential spectrum, respectively, is that $ {K_e}$ be a compact subset of $ K$ which contains the boundary of $ K$. Additionally, results are obtained on the question of which index values can be specified on the various components of the complement of $ {K_e}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0964860-2
PII: S 0002-9939(1988)0964860-2
Article copyright: © Copyright 1988 American Mathematical Society