Linear preservers of rank k projections

Paper

Abstract

Let

be a complex Hilbert space and

the real vector space of all self-adjoint finite rank bounded operators on

. We generalize the famous Wigner's theorem by characterizing linear maps on

which preserve the set of all rank

projections. In order to do this, we first characterize linear maps on the real vector space

of trace zero

hermitian matrices which preserve the subset of unitary matrices in

. We also study linear maps from

sending projections of rank

to finite rank projections. We prove some properties of such maps, e.g. that they send rank

projections to projections of a fixed rank. We give the complete description of such maps in the case

. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated.