Geometry of the left action of the p-Schatten groups.

MARÍA EUGENIA DI IORIO Y LUCERO

BANACH JOURNAL OF MATHEMATICAL ANALYSIS

BANACH MATHEMATICAL RESEARCH GROUP

Año: 2012

1735-8787

Abstract. Let H be an innite dimentional Hilbert space, Bp (H) the p-Schatten class of H and Up (H) be the Banach-Lie group of unitary operatorswhich are p-Schatten perturbations of the identity. Let A be a bounded self-adjoint operator in H. We show thatOA := UA : U 2 Up (H)is a smooth submanifold of the afinne space A + Bp (H) if only if A has closedrange. Futhermore, it is a homogeneous reductive space of Up (H). We intro-duce two metrics: one via the ambient Finsler metric induced as a submanifoldof A + Bp (H), the other, by means of the quotient Finsler metric providedby the homogeneous space structure. We show that OA is a complete metricspace with the rectiable distance of these metrics.