Genericity of Certain Classes of Unitary and Self-Adjoint Operators
Canadian mathematical bulletin, Tome 41 (1998) no. 2, pp. 137-139
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Choksi, J. R.; Nadkarni, M. G. Genericity of Certain Classes of Unitary and Self-Adjoint Operators. Canadian mathematical bulletin, Tome 41 (1998) no. 2, pp. 137-139. doi: 10.4153/CMB-1998-021-4
@article{10_4153_CMB_1998_021_4,
author = {Choksi, J. R. and Nadkarni, M. G.},
title = {Genericity of {Certain} {Classes} of {Unitary} and {Self-Adjoint} {Operators}},
journal = {Canadian mathematical bulletin},
pages = {137--139},
year = {1998},
volume = {41},
number = {2},
doi = {10.4153/CMB-1998-021-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-021-4/}
}
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[1] 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] 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] 3. Simon, B., Operators with singular continuous spectrum: I. General operators. Ann. of Math. 141 (1995), 131–145. Google Scholar
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