Geometry of unitary orbits of pinching operators

Abstract

Let I I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H H . Let View the MathML source {pi}w1 (1≤w≤∞) (1≤w≤∞) be a family of mutually orthogonal projections on H H . The pinching operator associated with the former family of projections is given by P:I⟶I,P(x)=∑wi=1pixpi. Let UI UI denote the Banach–Lie group of the unitary operators whose difference with the identity belongs to I I . We study geometric properties of the orbit UI(P)={LuPLu∗:u∈UI}, UI(P)={LuPLu∗:u∈UI}, where Lu Lu is the left representation of UI UI on the algebra B(I) 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.