On the spectral picture of an irreducible subnormal operator
Author:
Paul McGuire
Journal:
Proc. Amer. Math. Soc. 104 (1988), 801808
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 of the plane is the spectrum of an irreducible subnormal operator if and only if has exactly one nontrivial Gleason part such that is the closure of . The main result of this paper is that the only additional requirement needed for the pair to be the spectrum and essential spectrum, respectively, is that be a compact subset of which contains the boundary of . Additionally, results are obtained on the question of which index values can be specified on the various components of the complement of .
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 J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 7594. MR 0068129 (16:835a)
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 K. Clancey and C. R. Putnam, The local spectral behavior of completely subnormal operators, Trans. Amer. Math. Soc. 163 (1972), 239244. MR 0291844 (45:934)
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 J. B. Conway, Subnormal operators, Research Notes in Math., vol. 51, Pitman, London, 1981. MR 634507 (83i:47030)
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 J. B. Conway, Spectral properties of certain operators on Hardy spaces of planar regions, preprint. MR 904484 (88h:47040)
 [5]
 R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. MR 0361893 (50:14335)
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 S. Fisher, Function theory on planar domains, Wiley, New York, 1983. MR 694693 (85d:30001)
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 T. W. Gamelin, Uniform algebras, Chelsea, New York, 1986.
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 D. Hadwin and E. Nordgren, The BergerShaw Theorem, Proc. Amer. Math. Soc. (to appear).
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 W. W. Hastings, The approximate point spectrum of a subnormal operator, J. Operator Theory 5 (1981), 119126. MR 613052 (82h:47021)
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 P. J. McGuire, On algebras generated by a subnormal operator, preprint; J. Functional Anal. 79 (1988).
 [11]
 R. F. Olin and J. E. Thomson, Irreducible operators whose spectra are spectral sets, Pacific J. Math. 91 (1980), 431434. MR 615690 (82j:47012)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809648602
PII:
S 00029939(1988)09648602
Article copyright:
© Copyright 1988
American Mathematical Society
