The compatible Grassmannian

Abstract

Let A be a positive injective operator in a Hilbert space (H,〈{dot operator},{dot operator}〉), and denote by [ {dot operator}, {dot operator} ] the inner product defined by A: [f, g] = 〈A f, g〉. A closed subspace S⊂H is called A-compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product [ {dot operator}, {dot operator} ]. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=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↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form [ {dot operator}, {dot operator} ]. A Banach-Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form [ {dot operator}, {dot operator} ]. 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 ≤ p ≤ ∞, in the presence of a fixed [ {dot operator}, {dot operator} ]-orthogonal (direct sum) decomposition of H, H=S0+N0, 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 [ {dot operator}, {dot operator} ]. It carries the locally transitive action of the Banach-Lie group of invertible operators which preserve [ {dot operator}, {dot operator} ], and are of the form G = 1 + K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means 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.