Manifolds of algebraic elements in the algebra L(H) of bounded linear operators

Abstract

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the algebraic structure of Z, a torsionfree affine connection $\nabla$ (that is invariant under the group G= Aut (Z) of automorphisms of Z) is defined on each of these connected components and the geodesics are computed. In case M consists of elements that have a fixed finite rank r, (0<r<\infty), G-invariant Riemann and K\"ahler structures are defined on M which in this way becomes a totally geodesic symmetric holomorphic manifold.