Abstract
In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.
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J. E. Martínez-Legaz has been supported by the MICINN of Spain, Grant MTM2011-29064-C03-01, by the Barcelona Graduate School of Economics and by the Government of Catalonia. He is affiliated to MOVE (Markets, Organizations and Votes in Economics).
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García Ramos, Y., Martínez-Legaz, J.E. & Simons, S. New results on q-positivity. Positivity 16, 543–563 (2012). https://doi.org/10.1007/s11117-012-0191-7
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DOI: https://doi.org/10.1007/s11117-012-0191-7
Keywords
- q-Positive sets
- Symmetrically self-dual spaces
- Monotonicity
- Symmetrically self-dual Banach spaces
- Lipschitz mappings