A non-surjective Wigner-type theorem by way of equal pairs of subspaces
Authors: Mark Pankov
Summary: Let H be an infinite-dimensional complicated Hilbert area and let G∞(H) be the set of all closed subspaces of H whose dimension and codimension each are infinite. We examine (not essentially surjective) transformations of G∞(H) sending each pair of subspaces to an equal pair of subspaces; two pairs of subspaces are equal if there’s a linear isometry sending one in all these pairs to the opposite. Let f be such a metamorphosis. We present that there’s a distinctive as much as a scalar a number of linear or conjugate linear isometry L:H→H such that for each X∈G∞(H) the picture f(X) is the sum of L(X) and a sure closed subspace O(X) orthogonal to the vary of L. Within the case when H is separable, we give the next adequate situation to claim that f is induced by a linear or conjugate linear isometry: if O(X)=0 for a sure X∈G∞(H), then the identical holds for all X∈G∞(H).
