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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0308827-4
PII: S 0002-9939(1972)0308827-4
Keywords: Polynomially bounded, absolutely continuous, Radon-Nikodym derivative, $ {H^p}$-space, spectral radius
Article copyright: © Copyright 1972 American Mathematical Society