Geometry of unitary orbits of pinching operators

Let II be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space HH. Let {pi}1w(1≤w≤∞)(1≤w≤∞) be a family of mutually orthogonal projections on HH. The pinching operator associated with the former family of projections is given by P:I⟶I,P(x)=∑i=1wpixpi. Let UIUI denote the Banach–Lie group of the unitary operators whose difference with the identity belongs to II. We study geometric properties of the orbit UI(P)={LuPLu∗:u∈UI},UI(P)={LuPLu∗:u∈UI}, where LuLu is the left representation of UIUI on the algebra B(I)B(I) of bounded operators acting on II. The results include necessary and sufficient conditions for UI(P)UI(P) to be a submanifold of B(I)B(I). Special features arise in the case of the ideal KK of compact operators. In general, UK(P)UK(P) turns out to be a non complemented submanifold of B(K)B(K). We find a necessary and sufficient condition for UK(P)UK(P) to have complemented tangent spaces in B(K)B(K). We also show that UI(P)UI(P) is a covering space of another orbit of pinching operators.