The structure and asymptotic behavior of polynomially compact operators
Author:
Frank Gilfeather
Journal:
Proc. Amer. Math. Soc. 25 (1970), 127-134
MSC:
Primary 47.40
DOI:
https://doi.org/10.1090/S0002-9939-1970-0257791-3
MathSciNet review:
0257791
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A. R. Bernstein and A. Robinson proved that every polynomially compact operator in Hilbert space has nontrivial invariant subspaces. This paper gives a structure theorem for these operators. We show that a polynomially compact operator is the finite sum of translates of operators which have the property that a finite power of the operator is compact. Furthermore, the spectrum of polynomially compact operators is completely described. Conditions are given to determine the weak and strong asymptotic behavior of a polynomially compact contraction in Hilbert space.
- Tsuyoshi Andô, On hyponormal operators, Proc. Amer. Math. Soc. 14 (1963), 290–291. MR 145353, DOI https://doi.org/10.1090/S0002-9939-1963-0145353-4
- William B. Arveson and Jacob Feldman, A note on invariant subspaces, Michigan Math. J. 15 (1968), 61–64. MR 223922
- Sterling K. Berberian, Introduction to Hilbert space, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1961. MR 0137976
- F. E. Browder and W. V. Petryshyn, The solution by iteration of linear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 566–570. MR 190744, DOI https://doi.org/10.1090/S0002-9904-1966-11543-4
- Frank Gilfeather, Asymptotic convergence of operators in Hilbert space, Proc. Amer. Math. Soc. 22 (1969), 69–76. MR 247508, DOI https://doi.org/10.1090/S0002-9939-1969-0247508-2 ---, The structure of non-unitary operators and their asymptotic behavior, Thesis, University of California, Irvine, Calif., 1969.
- T. A. Gillespie, An invariant subspace theorem of J. Feldman, Pacific J. Math. 26 (1968), 67–72. MR 231232
- P. R. Halmos, Invariant subspaces of polynomially compact operators, Pacific J. Math. 16 (1966), 433–437. MR 193505
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
- Joseph G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458. MR 149282
- J. G. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476. MR 173161, DOI https://doi.org/10.1090/S0002-9947-1965-0173161-3
- Béla Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967 (French). MR 0225183
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47.40
Retrieve articles in all journals with MSC: 47.40
Additional Information
Keywords:
Polynomially compact operator,
asymptotic behavior,
structure theorem,
determination of spectrum,
invariant subspaces,
hyponormal operator,
normal operator
Article copyright:
© Copyright 1970
American Mathematical Society