The compatible Grassmanian

E. ANDRUCHOW, E. CHIUMIENTO ,M. E. DI IORIO Y LUCERO

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2014 vol. 32 p. 1 - 27

0926-2245

Let A be a positive injective operator in a Hilbert space (H,< , >), and denote by [ , ] the  inner product defined by A: [f,g]=. A closed subspace S of H is called  A-compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product [ , ]. Equivalently, if there exists a necessarily unique idempotent operator Q_S such that R(Q_S)=S, which is symmetric for this inner product. The compatible Grassmannian Gr_A is the set of all A-compatible subspaces of H. By parametrizing it via the one to one correspondence S--> Q_S, this set is shown to be a differentiable submanifold of the Banach space of all operators in H which are symmetric with respect to the form [ , ]. A Banach-Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form [ , ].  Each connected component in Gr_A of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie-Poisson space. For 1 leq p leq infty, in the presence of a fixed  [ , ]-orthogonal decomposition of H, H=S_0 + N_0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is  shown to be a submanifold of the Banach space of p-Schatten operators which are symmetric for the form [ , ]. It carries the locally transitive action of the Banach-Lie group of invertible operators which preserve [ , ], and are of the form G=1+K, with K in p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.