Monotone and convex operator functions

Abstract

Introduction.Monotone matrix functions of arbitrarily high order were introduced by Charles Loewner in the year 1934 [9](3) while studying realvalued functions which are analytic in their domain of definition, and continued in the complex domain, are regular in the entire open upper half-plane with non-negative imaginary part. The monotone matrix functions of order re defined later in the introduction deal with functions of matrices whose independent and dependent variables are real symmetric matrices of order re; monotone of arbitrarily high order means monotone for each finite integer re. The class corresponding to ra = 1 represents the functions which are monotone in the ordinary sense. Monotone operator functions are precisely the class of monotone matrix functions of arbitrarily high order. Loewner showed that analyticity plus the property of mapping the complex open upper halfplane into itself is characteristic for the class of monotone matrix functions of arbitrarily high order. Literature on this subject and its by-products, convex matrix functions and matrix functions of bounded variation, consists of but four papers, two by Loewner [9; 10 ] and one each from two of his doctoral students, F. Krauss [8] and O. Dobsch [5].The principal objective in this work was to discover whether convex operator functions (i.e. convex matrix functions of arbitrarily high order) satisfy properties analogous to those known for monotone operator functions. From the classical case it might be thought if a function is convex, its derivative should be monotone.

Loewner matrices and operator convexity

“…See [17] Corollary 2.6. This, in turn implies that f is operator convex, by a theorem of Bendat and Sherman [7]; see also [17] Theorem 2.4.…”

“…Assume that f is a C 2 function from (0, ∞) into itself, f (0) = 0 and f (0) = 0. We say that f is operator convex if Following Loewner's seminal work there have been several studies of these two classes of functions; see in particular [1,7,8,13,[15][16][17]19,20]. The emphasis of the present paper is on Loewner matrices, their spectral properties, and their role in characterising operator convexity.…”

The Calderón problem for Hilbert couples

“…Simplifications and new proofs of this theorem have been provided by several authors, cf. [2], [14], [20], [28], [13], cf. also Donoghue's book [10] which contains a fairly comprehensive exposition of the theory up until 1974.…”

Об Одном Классе Операторно-Монотонных Функций Нескольких Переменных

“…Классическая теорема Лёвнера [1]- [3] (см. также [4; замечания к § X.2]) утверждает, что непрерывная функция : (0, +∞) → R обладает свойством ( ) ( ) для всех пар ограниченных самосопряженных операторов , в гильбертовом пространстве таких, что 0 < , тогда и только тогда, когда аналитически продолжается в область C ∖ (−∞, 0] и это продолжение отображает открытую верхнюю полуплоскость в свое замы-кание.…”