Geometry of unitary orbits of pinching operators

TL;DR

This paper explores the geometric structure of unitary orbits of pinching operators on symmetrically-normed ideals, establishing conditions for submanifold properties, complemented tangent spaces, and analyzing the orbit's topology with applications to compact normal operators.

Contribution

It provides new geometric insights into the structure of unitary orbits of pinching operators, including submanifold criteria and topological properties, extending understanding in operator theory.

Findings

UI(P) is a submanifold under certain conditions.

UK(P) can be non-complemented in B(K).

UI(P) is a covering space of another orbit.

Abstract

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let ${p_i}_1 ^w$ $(1\leq w \leq \infty)$ 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)=\sum_{i=1}^{w} p_i x p_i. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study several geometric properties of the orbit UI(P)={L_{u} P L_{u^*} : u \in UI}, where L_u is the left representation of UI on the algebra 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 natural orbit of P. A quotient Finsler metric is introduced, and the induced rectifiable is studied. In addition, we give an application of the results on UI(P) to the topology of the UI-unitary orbit of a compact normal operator.