Contractions without cyclic vectors
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- by Béla Sz.-Nagy and Ciprian Foiaş PDF
- Proc. Amer. Math. Soc. 87 (1983), 671-674 Request permission
Abstract:
It is proved that if $T$ is a completely nonunitary contraction on Hilbert space such that ${T^{*n}}$ does not converge strongly to 0 as $n \to \infty$, there is an integer $N > 0$ so that none of the powers ${T^{*m}}$ with $m \geqslant N$ has a cyclic vector. Both conditions on $T$ are essential, and the integer $N$ is not universal, i.e., it depends on $T$.References
- Charles A. Berger, Intertwined operators and the Pincus principal function, Integral Equations Operator Theory 4 (1981), no. 1, 1–9. MR 602617, DOI 10.1007/BF01682744
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- Béla Sz.-Nagy and Ciprian Foiaş, Vecteurs cycliques et commutativité des commutants, Acta Sci. Math. (Szeged) 32 (1971), 177–183 (French). MR 305117
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 671-674
- MSC: Primary 47A15; Secondary 47A20
- DOI: https://doi.org/10.1090/S0002-9939-1983-0687638-1
- MathSciNet review: 687638