Lagrangian Grassmannian in infinite dimension
| Autor: | Esteban Andruchow, Gabriel Larotonda |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2009 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Geodesic Matemáticas LAGRANGIAN SUBSPACE General Physics and Astronomy Lagrangian Grassmannian Matemática Pura purl.org/becyt/ford/1 [https] symbols.namesake SYMPLECTIC GEOMETRY Unitary group FOS: Mathematics Operator Algebras (math.OA) Mathematical Physics Mathematics REAL AND COMPLEX DIFFERENTIAL GEOMETRY COMPLEX STRUCTURE ANALYSIS ON MANIFOLDS Mathematical analysis Hilbert space Mathematics - Operator Algebras purl.org/becyt/ford/1.1 [https] Linear subspace 53D12 (Primary) 58B20 (Secondary) GLOBAL ANALYSIS Differential Geometry (math.DG) SHORT GEODESIC Homogeneous space symbols Orthogonal group Geometry and Topology Operator norm CIENCIAS NATURALES Y EXACTAS |
| Zdroj: | CONICET Digital (CONICET) Consejo Nacional de Investigaciones Científicas y Técnicas instacron:CONICET |
| 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: | OpenAIRE |
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