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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.

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert spacebe a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given bypixpi.Let U I denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study several geometric properties of the orbitwhere Lu is the left representation of U I on the algebra B(I) of bounded operators acting on I. The results include necessary and sufficient conditions for U I (P ) to be a submanifold of B(I). Special features arise in the case of the ideal K of compact operators. In general, U K (P ) turns out to be a non complemented submanifold of B(K). We find a necessary and sufficient condition for U K (P ) to have complemented tangent spaces in B(K). We also show that U I (P ) is a covering space of another natural orbit of P . A quotient Finsler metric is introduced, and the induced rectifiable is studied. In addition, we give an application of the results on U I (P ) to the topology of the U I -unitary orbit of a compact normal operator.