Isometries of Grassmann spaces, II

Abstract

Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set P∞(H) of all projections of infinite rank and infinite corank when H is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian P∞(H). However, the same method gives an alternative proof in the case of finite rank projections.