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

Let H be an infinite-dimensional complex Hilbert space and let G∞(H) be the set of all closed subspaces of H whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of G∞(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→ H such that for every X∈ G∞(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∈ G∞(H), then the same holds for all X∈ G∞(H).