Orbits in symmetric spaces, II

TL;DR

This paper characterizes when the orbit of an element in symmetric Banach spaces is the closed convex hull of its extreme points and applies this to ideals of compact operators.

Contribution

It provides necessary and sufficient conditions for the convex hull property of orbits in symmetric spaces and extends results to operator ideals.

Findings

Characterization of orbit convex hull properties in symmetric spaces

Conditions for elements in Banach function and sequence spaces

Application to symmetrically normed ideals of compact operators

Abstract

Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or $(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.