Fixed points of polarity type operators

TL;DR

This paper generalizes the fixed point problem of the polarity operator in Hilbert spaces, showing solutions are ellipsoids when the linear operator is positive definite, and exploring related convex analysis results.

Contribution

It extends the fixed point analysis of polarity operators to include linear transformations, characterizing solutions as ellipsoids under positive definiteness.

Findings

Unique ellipsoid solutions for positive definite operators

Multiple or no solutions for non-positive definite operators

Connections to convex analysis and Hilbert space characterization

Abstract

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 $\mathbb{R}^n$, 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 equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possible independent interest, 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.