In this paper we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

中文翻译：

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关于实算子单调函数的注解

在本文中，我们开始研究算子元组函数的实算子单调性，这些函数是具有函数演算的多元结构映射，称为自由函数，保留有界线性希尔伯特空间算子的实部（或厄米部分）之间的顺序。我们根据自伴随域上的普通算子单调自由函数来完全表征开放凸自由域上的此类函数。进一步假设更严格的自由全纯性，我们证明所有这些函数都是具有完全正非常数部分的仿射线性。这个问题是由 David Blecher 在 2016 年 12 月在班加罗尔举行的两年一度的 OTOA 会议上提出的。