The compatible Grassmannian

TL;DR

This paper studies the geometry and structure of the compatible Grassmannian associated with a positive operator in a Hilbert space, revealing its manifold properties, symplectic leaves, and group actions, along with a restricted version related to Schatten classes.

Contribution

It introduces the compatible Grassmannian as a differentiable manifold, explores its symplectic structure, and analyzes a restricted version with Schatten class operators, including group actions and index characterizations.

Findings

The compatible Grassmannian is a differentiable submanifold.

Connected components are symplectic leaves in a Banach Lie-Poisson space.

Restricted Grassmannian characterized by Fredholm index of projection pairs.

Abstract

Let $A$ be a positive injective operator in a Hilbert space (\h, <,>), and denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace $\s \subset \h$ 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 $\s\leftrightarrow Q_\s$, 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 $1\le p \le \infty$, in the presence of a fixed [,]-orthogonal decomposition of $\h$, $\h=\s_0\dot{+} \n_0$, 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.