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

DOI:
https://doi.org/10.1090/S0002-9939-1988-0964860-2

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 .

**[1]**Joseph Bram,*Subnormal operators*, Duke Math. J.**22**(1955), 75–94. MR**0068129****[2]**K. F. Clancey and C. R. Putnam,*The local spectral behavior of completely subnormal operators*, Trans. Amer. Math. Soc.**163**(1972), 239–244. MR**0291844**, https://doi.org/10.1090/S0002-9947-1972-0291844-5**[3]**John B. Conway,*Subnormal operators*, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR**634507****[4]**John B. Conway,*Spectral properties of certain operators on Hardy spaces of planar regions*, Integral Equations Operator Theory**10**(1987), no. 5, 659–706. Toeplitz lectures 1987 (Tel-Aviv, 1987). MR**904484**, https://doi.org/10.1007/BF01195796**[5]**Ronald G. Douglas,*Banach algebra techniques in operator theory*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR**0361893****[6]**Stephen D. Fisher,*Function theory on planar domains*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR**694693****[7]**T. W. Gamelin,*Uniform algebras*, Chelsea, New York, 1986.**[8]**D. Hadwin and E. Nordgren,*The Berger-Shaw Theorem*, Proc. Amer. Math. Soc. (to appear).**[9]**William W. Hastings,*The approximate point spectrum of a subnormal operator*, J. Operator Theory**5**(1981), no. 1, 119–126. MR**613052****[10]**P. J. McGuire,*On**-algebras generated by a subnormal operator*, preprint; J. Functional Anal.**79**(1988).**[11]**Robert F. Olin and James E. Thomson,*Irreducible operators whose spectra are spectral sets*, Pacific J. Math.**91**(1980), no. 2, 431–434. MR**615690**

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0964860-2

Article copyright:
© Copyright 1988
American Mathematical Society