Geometry of the left action of the p-Schatten groups

Let H be an infinite dimensional Hilbert space, [B.sub.p] (H) the p-Schatten class of H and [U.sub.p](H) be the Banach-Lie group of unitary operators which are p-Schatten perturbations of the identity. Let A be a bounded self-adjoint operator in H. We show that [O.sub.A]:= {UA: U [member of] [U.sub.p](H)} is a smooth submanifold of the affine space A + [B.sub.p](H) if only if A has closed range. Furthermore, it is a homogeneous reductive space of [U.sub.p](H). We introduce two metrics: one via the ambient Pinsler metric induced as a submanifold of A + [B.sub.p](H), the other, by means of the quotient Pinsler metric provided by the homogeneous space structure. We show that [O.sub.A] is a complete metric space with the rectifiable distance of these metrics. Key words and phrases. Analytic submanifold, Pinsler metric, Riemannian metric, Schatten operator.