Abstract
We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ+ into ℝ+ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.
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Original Russian Text © Dinh Trung Hoa and O.E. Tikhonov, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 3, pp. 9–14.
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Hoa, D.T., Tikhonov, O.E. To the theory of operator monotone and operator convex functions. Russ Math. 54, 7–11 (2010). https://doi.org/10.3103/S1066369X10030023
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DOI: https://doi.org/10.3103/S1066369X10030023