A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces

Abstract

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of ${\mathcal G}_{\infty}(H)$ sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let $f$ be such a transformation. We show that there is a unique up to a scalar multiple linear or conjugate linear isometry $L:H\to H$ such that for every $X\in {\mathcal G}_{\infty}(H)$ the image $f(X)$ is the sum of $L(X)$ and a certain closed subspace $O(X)$ orthogonal to the range of $L$. In the case when $H$ is separable, we give the following sufficient condition to assert that $f$ is induced by a linear or conjugate linear isometry: if $O(X)=0$ for a certain $X\in {\mathcal G}_{\infty}(H)$, then the same holds for all $X\in {\mathcal G}_{\infty}(H)$.