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Adjointable maps between linear orthosets

Jan Paseka, Thomas Vetterlein

TL;DR

The work advances a framework that encodes Hermitian-space structure in linear orthosets through the orthogonality relation and a zero element, and studies adjointable maps as the bridge between the two worlds. It proves that under mild hypotheses, adjointable maps between linear orthosets come from quasilinear Hermitian maps, and linearity yields adjointability in the usual sense, linking to Wigner-type results that orthoisomorphisms are realized by quasiunitary maps for sufficiently rich spaces. It further develops an orthomodular–Dacey correspondence, showing how adjointable inclusions characterize orthomodular spaces and, under transitivity, recover Solér’s theorem for real, complex, and quaternionic Hilbert spaces. The paper also extends these ideas to partial correspondences, establishing a robust theory of partial quasiisometries and partial orthometries that generalizes the full adjointable correspondence to subspace pairs. Together, these results illuminate how orthogonality-based structures determine the underlying Hilbert-space geometry and its linear maps, enabling reconstruction and classification in both finite and infinite dimensions.

Abstract

Given an (anisotropic) Hermitian space $H$, the collection $P(H)$ of at most one-dimensional subspaces of $H$, equipped with the orthogonal relation $\perp$ and the zero linear subspace $\{0\}$, is a linear orthoset and up to orthoisomorphism any linear orthoset of rank $\geq 4$ arises in this way. We investigate in this paper the correspondence of structure-preserving maps between Hermitian spaces on the one hand and between the associated linear orthosets on the other hand. Our particular focus is on adjointable maps. We show that, under a mild assumption, adjointable maps between linear orthosets are induced by quasilinear maps between Hermitian spaces and if the latter are linear, they are adjointable as well. Specialised versions of this correlation lead to Wigner-type theorems; we see, for instance, that orthoisomorphisms between the orthosets associated with at least $3$-dimensional Hermitian spaces are induced by quasiunitary maps. In addition, we point out that orthomodular spaces of dimension $\geq 4$ can be characterised as irreducible Fréchet orthosets such that the inclusion map of any subspace is adjointable. Together with a transitivity condition, we may in this way describe the infinite-dimensional classical Hilbert spaces.

Adjointable maps between linear orthosets

TL;DR

The work advances a framework that encodes Hermitian-space structure in linear orthosets through the orthogonality relation and a zero element, and studies adjointable maps as the bridge between the two worlds. It proves that under mild hypotheses, adjointable maps between linear orthosets come from quasilinear Hermitian maps, and linearity yields adjointability in the usual sense, linking to Wigner-type results that orthoisomorphisms are realized by quasiunitary maps for sufficiently rich spaces. It further develops an orthomodular–Dacey correspondence, showing how adjointable inclusions characterize orthomodular spaces and, under transitivity, recover Solér’s theorem for real, complex, and quaternionic Hilbert spaces. The paper also extends these ideas to partial correspondences, establishing a robust theory of partial quasiisometries and partial orthometries that generalizes the full adjointable correspondence to subspace pairs. Together, these results illuminate how orthogonality-based structures determine the underlying Hilbert-space geometry and its linear maps, enabling reconstruction and classification in both finite and infinite dimensions.

Abstract

Given an (anisotropic) Hermitian space , the collection of at most one-dimensional subspaces of , equipped with the orthogonal relation and the zero linear subspace , is a linear orthoset and up to orthoisomorphism any linear orthoset of rank arises in this way. We investigate in this paper the correspondence of structure-preserving maps between Hermitian spaces on the one hand and between the associated linear orthosets on the other hand. Our particular focus is on adjointable maps. We show that, under a mild assumption, adjointable maps between linear orthosets are induced by quasilinear maps between Hermitian spaces and if the latter are linear, they are adjointable as well. Specialised versions of this correlation lead to Wigner-type theorems; we see, for instance, that orthoisomorphisms between the orthosets associated with at least -dimensional Hermitian spaces are induced by quasiunitary maps. In addition, we point out that orthomodular spaces of dimension can be characterised as irreducible Fréchet orthosets such that the inclusion map of any subspace is adjointable. Together with a transitivity condition, we may in this way describe the infinite-dimensional classical Hilbert spaces.
Paper Structure (6 sections, 46 theorems, 18 equations)

This paper contains 6 sections, 46 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Linear and projective spaces
  3. Hermitian spaces and linear orthosets
  4. Orthometric correspondences
  5. Orthomodular spaces and linear Dacey spaces
  6. Partial orthometric correspondences

Key Result

Theorem 2.4

Let $V$ be a linear space over a sfield $F$. We define and for $u, v \in V$, we put Then $\mathbf P(V)$, equipped with $\star$ and the zero subspace $\{0\}$, is a projective space. The dimension of $V$ coincides with the rank of $\mathbf P(V)$. Conversely, let $P$ be a projective space of rank $\geqslant 4$. Then there is a linear space $V$ and a projective isomorphism between $P$

Theorems & Definitions (108)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 98 more