Paper detail

Geometry of unitary orbits of pinching operators

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let ${p_i}_1 ^w$ $(1\leq w \leq \infty)$ be a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given by P: I --> I, P(x)=\sum_{i=1}^{w} p_i x p_i. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study several geometric properties of the orbit UI(P)={L_{u} P L_{u^*} : u \in UI}, where L_u is the left representation of UI on the algebra B(I) of bounded operators acting on I. The results include necessary and sufficient conditions for UI(P) to be a submanifold of B(I). Special features arise in the case of the ideal K of compact operators. In general, UK(P) turns out to be a non complemented submanifold of B(K). We find a necessary and sufficient condition for UK(P) to have complemented tangent spaces in B(K). We also show that UI(P) is a covering space of another natural orbit of P. A quotient Finsler metric is introduced, and the induced rectifiable is studied. In addition, we give an application of the results on UI(P) to the topology of the UI-unitary orbit of a compact normal operator.