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Products of Cesàro convergent sequences with applications to convex solid sets and integral operators


Author: Anton R. Schep
Journal: Proc. Amer. Math. Soc. 137 (2009), 579-584
MSC (2000): Primary 40G05, 46E30, 47B34
DOI: https://doi.org/10.1090/S0002-9939-08-09662-7
Published electronically: August 19, 2008
MathSciNet review: 2448578
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Abstract: Let $0\le a_{n}, b_{n}, c_{n}$ such that $a_{n}=b_{n}c_{n}$. If $a=\lim _{n\to \infty }a_{n}$, and $\{b_{n}\}$ and $\{c_{n}\}$ Cesàro converge to $b$, respectively $c$, then $a\le bc$. This implies that if in addition $\{b_{n}\}$ and $\{c_{n}\}$ are similarly ordered, then $a=bc$. As applications we prove that the pointwise product of two convex solid sets closed in measure is again closed in measure and a factorization result for kernels of regular integral operators on $L_{p}$–spaces.


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Additional Information

Anton R. Schep
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
MR Author ID: 155835
Email: schep@math.sc.edu

Keywords: Cesàro convergence, convex solid sets, integral operators
Received by editor(s): January 23, 2008
Published electronically: August 19, 2008
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.