2020/10/18 by Müller, Vladimir, Tomilov, Yuri
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2010.09129
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.