The compatible Grassmannian

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 ⊂ H is called Acompatible 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 ≤ p ≤ ∞, 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.2010 MSC: 58B10, 58B20, 47B10.

Proper subspaces and compatibility

“…Therefore, we obtain that C S,Tn − C S,T = G n P S//T G −1 n + (G + n ) −1 P + S//T G + n − P S//T − P + S//T → 0. This completes the proof of the continuity of the map defined in (3). ii) We set T 1 = G(T ), S 1 = G(S) and Q = P S//T .…”

“…We shall call them unitarizable operators. In the special case when E = H is a Hilbert space, these were studied in [4] and [3]. They can be obtained, for instance, as exponentials A = e iX , with X a symmetrizable operator.…”

On the geometry of normal projections in Krein spaces

“…This kind of results can be seen as a contribution to the differential geometry of projections, which has been a subject of study in different settings, see e.g. [10,11,20,8,4,3].…”

“…If this map takes values in U J , it will clearly be the required continuous local cross section for p E0 . The following argument to show that s(E) ∈ U J is borrowed and adapted from [3,Proposition 4.4]. It is useful to change from projections to symmetries via the map E → R E = 2E − I.…”

Operators which preserve a positive definite inner product

“…A closed linear subspace S ⊂ H is called compatible with A, A-compatible, or shortly compatible, if it admits a supplement which is orthogonal with respect to the inner product defined by A. In [4], the compatible Grassmannian was studied, namely Gr A = {S ⊂ H : S is closed and compatible with A}.…”