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ON CONNECTIONS BETWEEN DELTA-CONVEX MAPPINGS AND CONVEX OPERATORS

Published online by Cambridge University Press:  25 January 2007

Libor Veselý  and
Luděk Zajíček
Libor Veselý
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy ([email protected])
Luděk Zajíček
Affiliation:
Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic ([email protected])
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Abstract

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We study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping $F:(a,b)\to Y$ is the difference of two continuous convex operators whenever $Y$ belongs to a large class of Banach lattices which includes all $L^{p}(\mu)$ spaces ($1\leq p\leq\infty$). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.

Keywords

Primary 47H99Secondary 28B15delta-convex (d.c.) mappingconvex operatorLipschitz conditionnormed linear spaceBanach latticeJordan decomposition
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 3 , October 2006 , pp. 739 - 751
Copyright
Copyright © Edinburgh Mathematical Society 2006