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Operator monotone functions and Löwner functions of several variables. (English) Zbl 1268.47025

Ann. Math. (2) 176, No. 3, 1783-1826 (2012); corrigendum ibid. 180, No. 1, 403-405 (2014).
For positive integers \(d\) and \(n\), let CSAM\(_n^d\) be the set of \(d\)-tuples of commuting self-adjoint \(n\times n\) matrices and let CSA\(^d\) denote the set of \(d\)-tuples of commuting self-adjoint operators acting on an infinite-dimensional separable Hilbert space. Let \(E\) be open in \(\mathbb{R}^d\) and \(f\) be a real-valued \(C^1\)-function on \(E\). Then \(f\) is said to be locally \(M_n\)-monotone on \(E\) if, whenever \(S\) is in CSAM\(_n^d\) with \(\sigma(S)\) consisting of \(n\) distinct points in \(E\) and \(S(t)\) is a \(C^1\)-curve in CSAM\(_n^d\) with \(S(0)=S\) and \(\frac{d}{dt}S(t)|_{t=0}\geq 0\), then \(\frac{d}{dt}f(S(t))|_{t=0}\) exists and is \(\geq 0\). If in the above definition, we replace all occurrences of CSAM\(_n^d\) by CSA\(^d\), then the function \(f\) is called locally operator monotone.
In [Math. Z. 38, 177–216 (1934; Zbl 0008.11301, JFM 49.0714.01)], K. Löwner completely characterized functions of one variable (\(d=1\)) that are matrix or operator monotone. In the paper under review, the authors generalize Löwner’s results to higher dimensions. Let \(E\) and \(f\) be as above. They show that \(f\) is locally \(M_n\)-monotone if and only if \(f\) belongs to the Löwner class \(\mathcal{L}^{d}_{n}(E)\). For operator monotone functions, they establish the equivalence of the following statements: (a) \(f\) is \(M_n\)-monotone for all \(n\geq 1\); (b) \(f\) is operator monotone; (c) \(f\) belongs to the Löwner class \(\mathcal{L}(E)\).
There are also notions of global matrix and operator monotone functions. However, the results for these functions are less complete. The authors successfully characterize rational functions of two variables that are operator monotone on rectangles in \(\mathbb{R}^2\), but the problem remains open for rational functions of more than two variables, or for non-rational functions.
The paper ends with a list of open questions.
Reviewer: Trieu Le (Toledo)

MSC:

47A63 Linear operator inequalities
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A60 Functional calculus for linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
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