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An example in the theory of $AC$-operators

Author(s): Ian Doust; T. A. Gillespie
Journal: Proc. Amer. Math. Soc. 129 (2001), 1453-1457.
MSC (2000): Primary 47B40
Posted: October 24, 2000
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Abstract | References | Similar articles | Additional information

Abstract:

$AC$-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact $AC$-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.

References:

[BG]
E. Berkson and T.A. Gillespie, Absolutely continuous functions of two variables and well-bounded operators, J. London Math. Soc (2) 30 (1984), 305-321. MR 86c:47044

[CD1]
Cheng Qingping and I. Doust, Well-bounded operators on nonreflexive Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 799-808. MR 96f:47065

[CD2]
Cheng Qingping and I. Doust, Compact well-bounded operators, Preprint.

[Dow]
H.R. Dowson, Spectral theory of linear operators, London Mathematical Society Monographs 12, Academic Press, London, 1978. MR 80c:47022

[DW]
I. Doust and B.L. Walden, Compact $AC$-operators, Studia Math. 117 (1996), 275-287. MR 97a:47047

[PS]
A. Pe\lczynski and I. Singer, On non-equivalent and conditional bases in Banach spaces, Studia Math. 25 (1964), 5-25. MR 31:3831

[Sm]
D.R. Smart, Conditionally convergent spectral expansion, J. Austral. Math. Soc. 1 (1960), 319-333. MR 23:A3462

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Additional Information:

Ian Doust
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: i.doust@unsw.edu.au

T. A. Gillespie
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh, EH9~3JZ, Scotland
Email: t.a.gillespie@edinburgh.ac.uk

DOI: 10.1090/S0002-9939-00-05733-6
PII: S 0002-9939(00)05733-6
Keywords: $AC$-operators, compact operators
Received by editor(s): August 27, 1999
Posted: October 24, 2000
Additional Notes: The work of the first author was supported by the Australian Research Council. The second author thanks the School of Mathematics, University of New South Wales for its hospitality when this work was undertaken.
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2000, American Mathematical Society

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