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Complex Uniform Convexity and Riesz Measures

Published online by Cambridge University Press:  20 November 2018

Gordon Blower  and
Thomas Ransford
Gordon Blower
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England, United Kingdom e-mail: g.blower@lancaster.ac.uk
Thomas Ransford
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, Québec, G1K 7P4 e-mail: Thomas.Ransford@mat.ulaval.ca
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Abstract

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The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue ${{L}^{p}}$ spaces and the von Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\text{PL}$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals ${{c}^{p}}$ are 2-uniformly $\text{PL}$-convex for $1\,\le \,p\,\le \,2$.

Keywords

46B2046L52subharmonic functionsBanach spacesSchatten trace ideals
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 225 - 245
Copyright
Copyright © Canadian Mathematical Society 2004

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