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In search of convexity: diagonals and numerical ranges
Author(s) -
Müller V.,
Tomilov Yu.
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12480
Subject(s) - mathematics , numerical range , convexity , diagonal , bounded function , hilbert space , operator (biology) , convex set , regular polygon , range (aeronautics) , diagonal matrix , pure mathematics , combinatorics , mathematical analysis , convex optimization , geometry , biochemistry , chemistry , materials science , repressor , transcription factor , financial economics , economics , composite material , gene
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.