Fixed points of polarity type operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:467

Article

Reem, Daniel¹  Reich, Simeon¹ 

[1] Technion Israel Inst Technol, Dept Math, IL-3200003 Haifa, Israel

关键词: Converse to the Lax-Milgrarn theorem;    Ellipsoid;    Fixed point;    Minkowski functional;    Polar set;    Positive definite operator;   

DOI  :  10.1016/j.jmaa.2018.07.057

来源: Elsevier

【 摘 要 】

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in R-n, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm must be Euclidean. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equall. Our analysis yields a few by-products of possible independent interetion can have several (possibly infinitely many) solutions or no solution at ast, among them results related to coercive bilinear forms (essentially a quantitative convex analytic converse to the celebrated Lax-Milgram theorem from partial differential equations) and a characterization of real Hilbertian spaces. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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