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 ≤ 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 = 1 w p i x p i . Let U I denote the Banach–Lie group of the unitary operators whose difference with the identity belongs to I . We study geometric properties of the orbit U I ( P ) = { L u P L u ∗ : u ∈ U I } , where L u is the left representation of U I on the algebra B ( I ) of bounded operators acting on I . The results include necessary and sufficient conditions for U I ( P ) to be a submanifold of B ( I ) . Special features arise in the case of the ideal K of compact operators. In general, U K ( P ) turns out to be a non complemented submanifold of B ( K ) . We find a necessary and sufficient condition for U K ( P ) to have complemented tangent spaces in B ( K ) . We also show that U I ( P ) is a covering space of another orbit of pinching operators.