Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators

TL;DR

This paper characterizes transformations on real Hilbert spaces that preserve parallelogram areas, solving an open problem and extending Wigner's theorem, with applications to preserver problems in quantum mechanics.

Contribution

It provides a complete description of area-preserving maps on real Hilbert spaces and addresses an open preserver problem in two dimensions, extending results to higher dimensions.

Findings

Characterization of area-preserving transformations on  spaces.

Solution to an open preserver problem in 2D.

Extension of Wigner's theorem to non-linear maps.

Abstract

In this paper first we describe all (not necessarily linear or bijective) transformations on $\mathbb{R}^d$ with $2\leq d<\infty$ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not necessarily linear) bijections on an arbitrary real Hilbert space that preserve the latter quantity. This answers a question raised by Rassias and Wagner, and it can be considered as a variant of the famous Wigner theorem on real Hilbert spaces which plays an important role in quantum mechanics. As a consequence, we solve a preserver problem of Moln\'ar and Timmermann which has remained open stubbornly only in the two-dimensional case. Finally this two-dimensional result will be applied in order to strengthen their theorem in higher dimensions.