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Publicationes Mathematicae Debrecen

Transformations on the set of all $n$-dimensional subspaces of a Hilbert space preserving orthogonality

Máté Győry

Published Jul 1, 2004• 24 citations
Publisher

Abstract

In our paper we generalize Uhlhorn's version of Wigner's famous unitary-antiunitary theorem by describing the transformations preserving the orthogonality between higher dimensional subspaces under certain conditions.
DOI
10.5486/pmd.2004.3208

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Transformations on the set of all nnn-dimensional subspaces of a Hilbert space preserving orthogonality
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.
Orthogonality preserving transformations on the set of nnn-dimensional subspaces of a Hilbert space
We characterize bijective transformations on the set of all nnn-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of 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 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.