  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 $X\in {\mathcal
G}_{\infty}(H)$.
