Abstract

In his recent book From Hahn-Banach to monotonicity (Springer-Verlag, Berlin, 2008), S. Simons has introduced the notion of SSD space to provide an abstract algebraic framework for the study of monotonicity. Graphs of (maximal) monotone operators appear to be (maximally) qq-positive sets in suitably defined SSD spaces. The richer concept of SSDB space involves also a Banach space structure. In this paper we prove that the analog of the Fitzpatrick function of a maximally qq-positive subset MM in a SSD space (B,⌊⋅,⋅⌋)\left( B,\left\lfloor \cdot,\cdot \right\rfloor \right) is the smallest convex representation of MM. As a consequence of this result it follows that, in the case of a SSDB space, the conjugate with respect to the pairing ⌊⋅,⋅⌋\left \lfloor \cdot, \cdot \right \rfloor of any convex representation of MM provides a convex representation of MM, too. We also give a new proof of a characterization of maximally qq-positive subsets of SSDB spaces in terms of such special representations

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Juan Enrique Martínez-Legaz

Dep. d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

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Suggested citation

J. E. Martínez-Legaz. “On Maximally q-Positive Sets.” Journal of Convex Analysis 16 (2009), No. 3&4, 891–898. https://doi.org/10.68381/jca16054

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.