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.

中文翻译：

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寻找凸性：对角线和数值范围

我们证明了有界希尔伯特空间算子的所有可能的常数对角线的集合总是凸的。这尤其回答了 Bourin (2003) 的一个悬而未决的问题。此外，我们表明通勤算子元组的联合数值范围通常不是凸的，这填补了文献中的空白。我们还证明了 Asplund-Ptak 数值范围（对于运算符对是凸的）通常对于运算符元组不是凸的。