Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators (Q465233)

Let \(E\) be a real Hilbert space and NEWLINE\[NEWLINE\blacklozenge(a,b):=\sqrt{|a|^2.|b|^2-\langle a, b\rangle^2}NEWLINE\]NEWLINE the area of the parallelogram spanned by two vectors \(a,~b\in E\). Here, \(\langle~.~,~.~\rangle\) denotes the scalar product of \(E\) and \(|.|\) its norm. In the paper under review, the author answers a question raised by \textit{T. M. Rassias} and \textit{P. Wagner} [Aequationes Math. 66, No. 1--2, 85--89 (2003; Zbl 1085.39022)] by describing all maps \(\phi\) from \(E\) into itself preserving the area of parallelograms spanned by two vectors in \(E\); i.e., NEWLINE\[NEWLINE\blacklozenge(\phi(a),\phi(b))=\blacklozenge(a,b)NEWLINE\]NEWLINE for all \(a,~b\in E\). He also characterizes all maps \(\phi\) on \(B_s({\mathcal H})\) which preserve the norm of the commutators, and solves a preserver problem of \textit{L. Molnár} and \textit{W. Timmermann} [Int. J. Theor. Phys. 50, No. 12, 3857--3863 (2011; Zbl 1243.81079)]. Here, \(B_s({\mathcal H})\) is the real vector-space of all linear bounded self-adjoint operators on a complex separable Hilbert space \({\mathcal H}\) and \(\|.\|\) is a unitarily invariant norm on \(B_s({\mathcal H})\).