Wigner's type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank

Let H be a complex Hilbert space whose dimension is not less than 3 and let F-s (H) be the real vector space formed by all self-adjoint operators of finite rank on H. For every non-zero natural k < dim H we denote by P-k(H) the set of all rank k projections. Let H' be other complex Hilbert space of dimension not less than 3 and let L : F-s(H) -> F-s(H') be a linear operator such that L(P-k(H)) subset of P-m(H') for some natural k, m and the restriction of L to P-k(H) is injective. If H = H' and k = m, then L is induced by a linear or conjugate-linear isometry of H to itself, except the case dim H = 2k when there is another one possibility (we get a classical Wigner's theorem if k = m = 1). If dim H >= 2k, then k <= m. The main result describes all linear operators L satisfying the above conditions under the assumptions that H is infinite-dimensional and for any P, Q is an element of P-k(H) the dimension of the intersection of the images of L(P) and L(Q) is not less than m - k. (C) 2019 Elsevier Inc. All rights reserved.