Outgoing monotone geodesics of standard subspaces

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.

💡 Research Summary

The paper “Outgoing monotone geodesics of standard subspaces” develops a comprehensive classification of monotone geodesics in the space (\operatorname{Stand}(H)) of standard subspaces of a complex Hilbert space (H). The authors begin by establishing a real‑valued version of the classical Lax–Phillips theorem. In the original complex setting, an “outgoing” triple ((E,E_{+},U)) (closed subspace (E_{+}\subseteq E) and a unitary one‑parameter group (U_{t})) is equivalent to a model where (E\cong L^{2}(\mathbb R,M)), (E_{+}\cong L^{2}(\mathbb R_{+},M)) and (U_{t}) acts as the shift ((S_{t}f)(x)=f(x-t)). The authors prove that the same normal form holds when (E) is a real Hilbert space, using a real Hilbert space (M) and an orthogonal map (\psi).

A Fourier‑transform argument yields a “momentum‑space” version: after complexifying (M) to (M_{\mathbb C}= \mathbb C\otimes_{\mathbb R}M), the outgoing triple is realized on the (\sharp)‑invariant subspace (L^{2}(\mathbb R,M_{\mathbb C})^{\sharp}) with the Hardy space (H^{2}(\mathbb C_{+},M_{\mathbb C})^{\sharp}) playing the role of the positive subspace.

The paper then introduces reflection positivity. A unitary involution (\theta) on (E) is called reflection positive if (\langle\xi,\theta\xi\rangle\ge0) for all (\xi\in E_{+}). When (\theta) also satisfies (\theta U_{t}=U_{-t}\theta) and (U_{t}E_{+}\subseteq E_{+}) for (t\ge0), the quadruple ((E,E_{+},U,\theta)) is an “outgoing reflection‑positive orthogonal (or unitary) one‑parameter group”.

The central structural result (Theorem 1.2.4) shows that such a quadruple is equivalent to the existence of a function
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