Manifolds of linear involutions

Abstract

The set of bounded linear involutions on a complex Banach space X is equipped with a Banach manifold structure and an affine connection compatible with its embedding into B ( X ). Geodesic lines are characterized. Moreover, if X is a Hilbert space and the topology of the self-adjoint part of the manifold is strengthened to be compatible with the Hilbert-Schmidt metric, these geodesics are identified as minimal arcs between pairs of self-adjoint involutions whose straight line distance is less than 2.