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New results on \(q\)-positivity. (English) Zbl 1334.47053

Summary: 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.

MSC:

47H05 Monotone operators and generalizations
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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