2018/04/22 by Mark Pankov, Pankov, Mark · 1 citation
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1804.08156
openalex publication_date 2018/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a complex Hilbert space whose dimension is not less than 3 and\nlet mathcal Fs(H) be the real vector space formed by all self-adjoint\noperators of finite rank on H. For every non-zero natural k<\dim H we\ndenote by mathcal Pk(H) the set of all rank k projections. Let H'\nbe other complex Hilbert space of dimension not less than 3 and let\nL: mathcal Fs(H)\→ mathcal Fs(H') be a linear operator such that\nL( mathcal Pk(H))\⊂ mathcal Pm(H') for some natural k,m and\nthe restriction of L to mathcal Pk(H) is injective. If H=H' and\nk=m, then L is induced by a linear or conjugate-linear isometry of H to\nitself, except the case \dim H=2k when there is another one possibility (we\nget a classical Wigner's theorem if k=m=1). If \dim H\≥ 2k, then k\≤ m.\nThe main result describes all linear operators L satisfying the above\nconditions under the assumptions that H is infinite-dimensional and for any\nP,Q\∈ mathcal Pk(H) the dimension of the intersection of the images of\nL(P) and L(Q) is not less than m-k.\n