  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.
