In search of convexity: diagonals and numerical ranges

TL;DR

This paper investigates the convexity properties of numerical ranges and diagonals of operators, proving convexity for certain sets and non-convexity for others, thus clarifying open questions in operator theory.

Contribution

It establishes the convexity of constant diagonals for bounded Hilbert space operators and demonstrates non-convexity of joint numerical ranges for operator tuples, addressing open problems.

Findings

Constant diagonals of bounded operators form convex sets.

Joint numerical range of commuting operator tuples is generally not convex.

Asplund-Ptak numerical range is not convex for tuples of operators.

Abstract

We show that the set of all possible constant diagonals of a bounded Hilbert space operator is always convex. This, in particular, answers an open question of J.-C. Bourin ($2003$). Moreover, we show that the joint numerical range of a commuting operator tuple is in general not convex, which fills a gap in the literature. We also prove that the Asplund-Ptak numerical range (which is convex for pairs of operators) is, in general, not convex for tuples of operators.