A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

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.

On the relation between the monotone Riemannian metrics on the space of Gibbs thermal states and the linear response theory

“…It should be stressed that the estimates, Eqs. ( 61), ( 68), (71) and the series of inequalities in Section VI, are exact and cannot be inferred from any perturbation theory.…”

“…By approximation arguments this definition is extend-able for operators on an infinite dimensional Hilbert space (for a more information about operator monotone functions, see e.g. [36,71]).…”

“…In according with the Kubo and Ando theory of operator means (see, [35], Chapter V in ref. [36], [71]) to every operator mean corresponds a unique operator monotone function f ∈ F op , and conversely. If f ∈ F op then the corresponding mean is given by Eq.…”

Convenient Tail Bounds for Sums of Random Tensors

“…where X 1 , X 2 are positive-definite tensors and t ∈ [0, 1]. The concavity of tensor logarithm can be derived from Hansen-Pedersen Characterizations, see [7].…”

Local characterizations for the matrix monotonicity and convexity of fixed order

“…Ever since Charles Loewner (then known as Karl Löwner) introduced matrix monotone functions in 1934 [12], this class has been characterized in various ways. See for example [2,8] for survey and recent progress. The famous theorem established in the Loewner's paper states that a function that is matrix monotone of all orders on an interval, extends to upper half-plane as a Pick-Nevanlinna function: an analytic function with non-negative imaginary part.…”