A Note on the Carath\'{e}odory Number of the Joint Numerical Range

TL;DR

This paper establishes upper bounds on the Carathéodory number of joint numerical ranges of bounded self-adjoint operators, revealing their convexity properties and how they differ from general sets.

Contribution

It extends classical convexity results by providing new bounds on the Carathéodory number for joint numerical ranges of operators.

Findings

Carathéodory number of joint numerical range ≤ d-1

Improved bound ≤ d-2 for Hilbert spaces of dimension ≥ 3

Joint numerical ranges are less non-convex than general sets

Abstract

We show that the Carath\'{e}odory number of the joint numerical range of $d$ many bounded self-adjoint operators is at most $d-1$, and even at most $d-2$ if the underlying Hilbert space has dimension at least $3$. This extension of the classical convexity results for numerical ranges shows that also joint numerical ranges are significantly less non-convex than general sets.