## Abstract

We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M_{+}) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U,J,M_{+}) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J, and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U,J,M_{+}), with U assumed unitary and J-self-adjoint, and M_{+} reflecting the choices of the associated J-positive and U-invariant subspaces. We further consider the wider setting when U is only assumed J-self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M_{+} will be realized as spaces G(X), graph of a second linear operator X, where the possibilities for such X, as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis.

Original language | English |
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Article number | 102411 |

Journal | Advances in Applied Mathematics |

Volume | 141 |

DOIs | |

State | Published - 1 Oct 2022 |

Externally published | Yes |

## Keywords

- Harmonic analysis
- Hilbert space
- J-self-adjoint
- Krein space
- Osterwalder-Schrader axioms
- Potapov-Ginzburg transform
- Realization of input/output systems
- Reflection positive
- Riccati equation
- Schur analysis

## ASJC Scopus subject areas

- Applied Mathematics