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arXiv · 2012.02063

A geometric approach to Wigner-type theorems

Mark Pankov·Thomas Vetterlein
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Abstract

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of ${\mathcal P}(H)$ to itself is induced by a linear or conjugate-linear isometry ($H$ is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

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Mark Pankov, Thomas Vetterlein. 2020-12-03. A geometric approach to Wigner-type theorems. https://arxiv.org/abs/2012.02063

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