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Proceedings of the American Mathematical Society

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Spectral radius of an absolutely continuous operator

Author: Arnold Lebow
Journal: Proc. Amer. Math. Soc. 36 (1972), 511-514
MSC: Primary 47A60; Secondary 47A10
MathSciNet review: 0308827
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Abstract: An operator T on a Hilbert space is said to be absolutely continuous if, for every pair of vectors (x, y) and every nonnegative integer n, $ \langle {T^n}x,y\rangle $ is the nth Fourier coefficient of an $ {L_1}(0,2\pi )$ function $ {F_{xy}}$:

$\displaystyle \langle {T^n}x,y\rangle = \frac{1}{{2\pi }}\int_0^{2\pi } {{F_{xy}}(\theta ){e^{ - in\theta }}d\theta .} $

The main result of this paper is that if $ {F_{xy}}$ is in $ \cup \{ {L_p}:p > 1\} $ for all x and y then T has spectral radius less than one.

References [Enhancements On Off] (What's this?)

  • [1] G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1960; English transl., Springer-Verlag, 1969. MR 24 #A411; MR 40 #1750.
  • [2] A. Lebow, A power-bounded operator that is not polynomially bounded, Michigan Math. J. 15 (1968), 397–399. MR 0236753
  • [3] Gian-Carlo Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472. MR 0112040
  • [4] Morris Schreiber, Absolutely continuous operators, Duke Math. J. 29 (1962), 175–190. MR 0147912

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Keywords: Polynomially bounded, absolutely continuous, Radon-Nikodym derivative, $ {H^p}$-space, spectral radius
Article copyright: © Copyright 1972 American Mathematical Society