Complex Uniform Convexity and Riesz Measure
Printed: Apr 2004
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
$\PL$-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 $\PL$-convex for
$1\leq p\leq 2$.
subharmonic functions, Banach spaces, Schatten trace ideals
46B20 - Geometry and structure of normed linear spaces
46L52 - Noncommutative function spaces