Transformations of the unitary group on a Hilbert space
This paper contains nine results on the subject of transformations of spaces of operators, all of them dealing with transformations of the unitary group \(\mathcal U\) of a separable complex and infinite dimensional Hilbert space \(\mathcal H\); two of the results deal with symmetries (self-adjoint unitaries), and due to the correspondence \(p\mapsto \epsilon_p=2p-1\) among symmetries and orthogonal projections, these projections play a significant role in the paper. What follows is a typical result of the paper: let \(\phi:\mathcal U\to \mathcal U\) be bijective map with \[ \phi(VW)=\phi(V)\phi(W)\, \text{for all} \, V,W\in\mathcal U, \] then there exists an unitary or antiunitary operator \(U\) in \(\mathcal H\) such that \(\phi(V)=UVU^*\) (\(V\in\mathcal U\)). No additional hypothesis on the continuity of \(\phi\) is required.
scientific article; zbMATH DE number 6009629
scientific article