Abstract
This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.
Highlights
This paper is an expository devoted to an important class of real-valued functions introduced by Lowner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions
A useful and important class of real-valued functions is the class of operator monotone functions
Such functions were introduced by Lowner in a seminal paper [1]. These functions are functions of Hermitian matrices/operators preserving order. He established a relationship between operator monotonicity, the positivity of matrix of divided differences, and an important class of analytic functions, namely, Pick functions
Summary
A useful and important class of real-valued functions is the class of operator monotone functions. He established a relationship between operator monotonicity, the positivity of matrix of divided differences, and an important class of analytic functions, namely, Pick functions This concept is closely related to operator convex/concave functions which was studied afterwards by Kraus in [2]. Operator monotone functions have applications in many areas, including functional analysis, mathematical physics, information theory, and electrical engineering; see, for example, [12,13,14,15] This concept plays major roles in the so-called Kubo-Ando theory of operator connections and operator means.
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