Geometry of unitary orbits of pinching operators

Abstract

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let {pi}1w(1≤w≤∞) 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)=∑i=1wpixpi. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study geometric properties of the orbit UI(P)={LuPLu*:u∈UI}, where Lu 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 orbit of pinching operators.