Outgoing monotone geodesics of standard subspaces

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 \(\textrm{Stand}(\mathcal {H})\) of standard subspaces on some complex Hilbert space \(\mathcal {H}\). 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 \(\textrm{Stand}(\mathcal {H})\) arise from the unitary one-parameter groups described in Borchers’ Theorem and provide explicit examples of monotone geodesics that are not of this type.