Outgoing monotone geodesics of standard subspaces

TL;DR

This paper characterizes outgoing monotone geodesics in standard subspaces of Hilbert spaces, linking them to positive Hankel operators and providing explicit classifications and examples.

Contribution

It introduces a normal form for outgoing monotone geodesics and connects them to positive Hankel operators, expanding understanding of their structure.

Findings

Classified outgoing reflection positive orthogonal one-parameter groups.

Provided explicit symbols for positive Hankel operators.

Identified which monotone geodesics originate from Borchers' unitary groups.

Abstract

We prove a real version of the Lax-Phillips Theorem and classify outgoing reflection positive orthogonal one-parameter groups. Using these results, we provide a normal form for outgoing monotone geodesics in the set Stand(H) of standard subspaces on some complex Hilbert space H. As the modular operators of a standard subspace are closely related to positive Hankel operators, our results are obtained by constructing some explicit symbols for positive Hankel operators. We also describe which of the monotone geodesics in Stand(H) arise from the unitary one-parameter groups described in Borchers' Theorem and provide explicit examples of monotone geodesics that are not of this type.