Spectral radius of an absolutely continuous operator
HTML articles powered by AMS MathViewer
- by Arnold Lebow PDF
- Proc. Amer. Math. Soc. 36 (1972), 511-514 Request permission
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 ${F_{xy}}$: \[ \langle {T^n}x,y\rangle = \frac {1}{{2\pi }}\int _0^{2\pi } {{F_{xy}}(\theta ){e^{ - in\theta }}d\theta .} \] 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
-
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.
- A. Lebow, A power-bounded operator that is not polynomially bounded, Michigan Math. J. 15 (1968), 397–399. MR 236753
- Gian-Carlo Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472. MR 112040, DOI 10.1002/cpa.3160130309
- Morris Schreiber, Absolutely continuous operators, Duke Math. J. 29 (1962), 175–190. MR 147912
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 36 (1972), 511-514
- MSC: Primary 47A60; Secondary 47A10
- DOI: https://doi.org/10.1090/S0002-9939-1972-0308827-4
- MathSciNet review: 0308827