On the structure of the similarity orbits of Jordan operators as analytic homogeneous manifolds

Abstract

For Jordan elementsJ in a topological algebraB with unite, an open groupB ⁻¹ of invertible elements and continuous inversion we consider the similarity orbitsS _G (J)={gJg ⁻¹:g∈G} (G the groupB ⁻¹⋂{e+c:c∈I},I⊂B a bilateral continuous embedded topological ideal). We construct rational local cross sections to the conjugation mapping$\pi ^J G \to S_G \left( J \right)\left( {\pi ^J \left( g \right) = gJg^{ - 1} } \right)$ and give to the orbitS _G (J) the local structure of a rational manifold. Of particular interest is the caseB=L(H) (bounded linear operators on a separable Hilbert spaceH),I=B, for which we obtain the following:

• 1. If for a Hilbert space operator there exist norm continuous local similarity cross sections, then these can be chosen to be rational, especially holomorphic or real analytic.

• 2. The similarity orbit of a nice Jordan operator is a rational (especially holomorphic or real analytic) submanifold ofL(H).