In search of convexity: diagonals and numerical ranges

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.

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