Operator Monotone Functions: Characterizations and Integral Representations

Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related fields. This concept is closely related to operator convex/concave functions. In this paper, we provide their important examples and characterizations in terms of matrix of divided differences. Various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen-Pedersen characterizations. Moreover, operator monotone functions on the nonnegative reals have special properties, namely, they admit integral representations with respect to suitable Borel measures.