Isometries of Grassmann spaces

Botelho, Jamison, and Moln\' ar have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Moln\' ar, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surectivity assumption. A variety of tools is used in order to achieve our goal, such as topological, geometrical and linear algebra techniques. The famous two projections theorem for two finite rank projections will be re-proven using linear algebraic methods. A theorem of Gy\"ory and the second author on orthogonality preservers on Grassmann spaces will be strengthened as well. This latter result will be obtained by using Chow's fundamental theorem of geometry of Grassmannians.