Lagrangian Grassmannian in Infinite Dimension
| Autor: | Andruchow, Esteban, Larotonda, Gabriel |
|---|---|
| Rok vydání: | 2008 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.geomphys.2008.11.004 |
| Popis: | 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. Comment: 23 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|