Stampacchia's property, self-duality and orthogonality relations

TL;DR

This paper demonstrates that the validity of Stampacchia's theorem on a Banach space implies the space is isomorphic to a Hilbert space, linking variational inequalities, self-duality, and orthogonality.

Contribution

It establishes a novel characterization of Hilbert spaces via Stampacchia's theorem and explores connections between self-duality, orthogonality, and the cosine of operators.

Findings

Banach space satisfying Stampacchia's theorem is Hilbertian

Self-dual Banach spaces are characterized through orthogonality relations

The cosine of a linear operator characterizes Hilbert space structure

Abstract

We show that if the conclusion of the well known Stampacchia Theorem, on variational inequalities, holds on a Banach space X, then X is isomorphic to a Hilbert space. Motivated by this we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between existing notions of orthogonality and self-duality. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize Hilbert space structure. Finally, we present some consequences of our results to quadratic forms and to evolution triples.