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An example in the theory of $AC$-operators


Authors: Ian Doust and T. A. Gillespie
Journal: Proc. Amer. Math. Soc. 129 (2001), 1453-1457
MSC (2000): Primary 47B40
DOI: https://doi.org/10.1090/S0002-9939-00-05733-6
Published electronically: October 24, 2000
MathSciNet review: 1814172
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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.


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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: https://doi.org/10.1090/S0002-9939-00-05733-6
Keywords: $AC$-operators, compact operators
Received by editor(s): August 27, 1999
Published electronically: 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
Article copyright: © Copyright 2000 American Mathematical Society