Eduardo Chiumiento
Infinite Grassmanians and operator theory
The Grassmannian of an infinite-dimensional Hilbert space is the set consisting of all the closed subspaces. From an operator theoretic perspective, it may be identified with the set of all orthogonal projections on the Hilbert space. Thus, its connected components become homogeneous spaces, which can be endowed with a Finsler metric by using the operator norm on tangent spaces. In these lectures we will present old and new results on the differential and metric properties of the Grassmannian of a Hilbert space. Then we will consider the geometric structure of other Grasssmannians obtained by imposing restrictions on the subspaces such as restricted Grassmannians, projections with compact commutator, etc. Also we will discuss the relation with other infinite dimensional manifolds such as manifolds of idempotents, manifolds of partial isometries and abstract homogeneous spaces. We will emphasize the interplay between geometric aspects of infinite Grassmnnians and operator theory. It is based on a joint work with Esteban Andruchow and Gustavo Corach.
