Lagrangian Grassmannian in Infinite Dimension

TL;DR

This paper explores the geometry of the Lagrangian Grassmannian in infinite-dimensional Hilbert spaces, establishing geodesic connectivity, a Finsler metric, and the validity of Hopf-Rinow theorem, extending to Schatten class groups.

Contribution

It introduces a geometric framework for the infinite-dimensional Lagrangian Grassmannian, including geodesic completeness and metric properties, extending classical finite-dimensional results.

Findings

Any two Lagrangian subspaces can be connected by a geodesic.

The Finsler metric satisfies the Hopf-Rinow theorem in this setting.

Results extend to Schatten class Banach-Lie groups.

Abstract

Given a complex structure $J$ on a real (finite or infinite dimensional) Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian $\Lambda(H)$ of $H$, i.e. the set of closed linear subspaces $L\subset H$ such that $$J(L)=L^\perp.$$ The complex unitary group $U(H_J)$, consisting of the elements of the orthogonal group of $H$ which are complex linear for the given complex structure, acts transitively on $\Lambda(H)$ and induces a natural linear connection in $\Lambda(H)$. It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced, if one regards subspaces $L$ as projections $p_L$ (=the orthogonal projection onto $L$) or symmetries $\e_L=2p_L-I$, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in $\Lambda(H)$: a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. We extend these results to the classical Banach-Lie groups of Schatten.