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Some unsolved problems of unknown depth about operators on Hilbert space

Published online by Cambridge University Press:  14 February 2012

P. R. Halmos
P. R. Halmos
Affiliation:
Department of Mathematics, University of California (Santa Barbara)
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Synopsis

The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 1 , 1976 , pp. 67 - 76
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

1Apostol, C., Foiaş, C. and Voiculescu, D.. Some results on non-quasitriangular operators IV. Rev. Roumaine Math. Pures Appl. 18 (1973), 487–514.Google Scholar
2Aronszajn, N. and Smith, K. T.. Invariant subspaces of completely continuous operators. Ann. of Math. 60 (1954), 345–350.CrossRefGoogle Scholar
3Arveson, W. B.. A density theorem for operator algebras. Duke Math. J. 34 (1967), 635–648.CrossRefGoogle Scholar
4Bastian, J. J. and Harrison, K. J.. Subnormal weighted shifts and asymptotic properties of normal operators. Proc. Amer. Math. Soc. 42 (1974), 475–479.CrossRefGoogle Scholar
5Berg, I. D.. On approximation of normal operators by weighted shifts, to appear.Google Scholar
6Bernstein, A. R. and Robinson, A.. Solution to an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 (1966), 421–431.CrossRefGoogle Scholar
7Brown, L. G.. Almost every proper isometry is a shift. Indiana Univ. Math. J. 23 (1973), 429–431.CrossRefGoogle Scholar
8Coburn, L. A.. Weyl's theorem for nonnormal operators. Michigan Math. J. 13 (1966), 285–288.CrossRefGoogle Scholar
9Coburn, L. A.. The C*-algebra generated by an isometry. II. Trans. Amer. Math. Soc. 137 (1969), 211–217.Google Scholar
10Devinatz, A. and Shinbrot, M.. General Wiener-Hopf operators. Trans. Amer. Math. Soc. 145 (1969), 467–469.CrossRefGoogle Scholar
11Fillmore, P. A., Stampfli, J. G. and Williams, J. P.. On the essential numerical range, the essential spectrum, and a problem of Halmos. Acta Sci. Math. (Szeged), 33 (1972), 179–192.Google Scholar
12Halmos, P. R.. Spectra and spectral manifolds. Ann. Soc. Polon. Math. 25 (1952), 43–49.Google Scholar
13Halmos, P. R.. Quasitriangular operators. Acta Sci. Math. (Szeged), 29 (1968), 283–293.Google Scholar
14Halmos, P. R.. Ten problems in Hilbert space. Bull. Amer. Math. Soc. 76(1970), 887–933Google Scholar
15Halmos, P. R.. Positive approximants of operators. Indiana Univ. Math. J. 21 (1972), 951–960.CrossRefGoogle Scholar
16Halmos, P. R.. Products of shifts. Duke Math. J. 39 (1972), 779–787.CrossRefGoogle Scholar
17Hartman, P. and Wintner, A.. On the spectra of Toeplitz's matrices. Amer. J. Math. 76 (1954), 867–882.CrossRefGoogle Scholar
18Pearcy, C. M. and Shields, A. L.. Almost commuting versus commuting finite matrices, to appear.Google Scholar
19Rogers, D. D.. Normal spectral approximation (Indiana Univ. Dissertation, 1975).Google Scholar
20Sarason, D. E.. Invariant subspaces and unstarred operator algebras. Pacific J. Math. 17 (1966), 511–517.CrossRefGoogle Scholar