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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mean-boundedness and Littlewood-Paley for separation-preserving operators


Authors: Earl Berkson and T. A. Gillespie
Journal: Trans. Amer. Math. Soc. 349 (1997), 1169-1189
MSC (1991): Primary 42A45, 42B25, 46E30, 47B40
MathSciNet review: 1407694
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Abstract: Suppose that $(\Omega ,\mathcal {M},\mu )$ is a $\sigma $-finite measure space, $1<p<\infty $, and $T: L^{p}(\mu )\to L^{p}(\mu )$ is a bounded, invertible, separation-preserving linear operator such that the linear modulus of $T$ is mean-bounded. We show that $T$ has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in $L^{p}(\mu )$ which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for $T$ is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra'' of $\mathbb {T}$, and to furnish $L^{p}(\mu )$ with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for $\ell ^{p}(\mathbb {Z})$.


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Additional Information

Earl Berkson
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, Illinois 61801
Email: berkson@symcom.math.uiuc.edu

T. A. Gillespie
Affiliation: Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland
Email: t.a.gillespie@edinburgh.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01896-5
PII: S 0002-9947(97)01896-5
Received by editor(s): August 17, 1995
Additional Notes: The work of the first author was supported by a grant from the National Science Foundation (U.S.A.).
Article copyright: © Copyright 1997 American Mathematical Society