Linear preservers of rank k projections

2025/12/11 by Plevnik, Lucijan
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2512.10645

Abstract

Let \mathcal H be a complex Hilbert space and \mathcal F_s (\mathcal H) the real vector space of all self-adjoint finite rank bounded operators on \mathcal H. We generalize the famous Wigner's theorem by characterizing linear maps on \mathcal F_s (\mathcal H) which preserve the set of all rank k projections. In order to do this, we first characterize linear maps on the real vector space \mathcal H_{0, 2k} of trace zero (2k) × (2k) hermitian matrices which preserve the subset of unitary matrices in \mathcal H_{0, 2k}. We also study linear maps from \mathcal F_s (\mathcal H) to \mathcal F_s (\mathcal K) sending projections of rank k to finite rank projections. We prove some properties of such maps, e.g. that they send rank k projections to projections of a fixed rank. We give the complete description of such maps in the case dim \mathcal H = 2. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated.

Citations

• A non-injective version of Wigner's theorem
• Wigner's type theorem in terms of linear operators which send\n projections of a fixed rank to projections of other fixed rank
• Wigner's theorem on Grassmann spaces

Related

• Wigner's type theorem in terms of linear operators which send\n projections of a fixed rank to projections of other fixed rank
• A non-injective version of Wigner's theorem
• Linear rank one preservers on tensor products of Hermitian matrices
• A geometric approach to Wigner-type theorems
• Positive maps which are bijective on the set of rank projections