In search of convexity: Diagonals and numerical ranges

0544892 - MÚ 2022 RIV GB eng J - Journal Article
Müller, Vladimír - Tomilov, Y.
In search of convexity: Diagonals and numerical ranges.
Bulletin of the London Mathematical Society. Roč. 53, č. 4 (2021), s. 1016-1029. ISSN 0024-6093. E-ISSN 1469-2120
R&D Projects: GA ČR(CZ) GX20-31529X
Institutional support: RVO:67985840
Keywords : convexity * Hilbert space operator
OECD category: Pure mathematics
Impact factor: 1.036, year: 2021
Method of publishing: Limited access
https://doi.org/10.1112/blms.12480

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 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.
Permanent Link: http://hdl.handle.net/11104/0321689