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Genericity of Certain Classes of Unitary and Self-Adjoint Operators

Published online by Cambridge University Press:  20 November 2018

J. R. Choksi  and
M. G. Nadkarni
J. R. Choksi
Affiliation:
Department of Mathematics & Statistics McGill University Montréal, Québec H3A 2K6, e-mail: choksi@math.mcgill.ca
M. G. Nadkarni
Affiliation:
Department of Mathematics University of Mumbai Vidyanagri Mumbai, 400 098 India, e-mail: nadkarni@mathbu.ernet.in
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Abstract

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In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, forma dense ${{G}_{\delta }}$. A similar theorem for bounded self-adjoint operators with a given normbound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added.

Keywords

47B15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 2 , 01 June 1998 , pp. 137 - 139
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Choksi, J. R. and Nadkarni, M. G. [CN], Baire category in spaces of measures, unitary operators and transformations. Invariant Subspaces and Allied Topics, (eds. H. Helson and B. S. Yadav), Narosa Publ. Co., New Delhi, 1990, 147–163.Google Scholar
2. del Rio, R., Jitomirskaya, S., Makarov, N. and Simon, B., Singular continuous spectrum is generic. Bull. Amer. Math. Soc. 31 (1994), 208–212.Google Scholar
3. Simon, B., Operators with singular continuous spectrum: I. General operators. Ann. of Math. 141 (1995), 131–145.Google Scholar