The calculus of operator functions and operator convexity

Abstract

IntroductionIf J is an open subinterval of R and f is a real-valued function defined on J then for each self-adjoint operator A on a finite-dimensional complex inner product space, the spectrum of which is contained in J, there is defined a self-adjoint operator which is denoted by f (A). One refers to the "operator function" f . If J and J ′ are open subintervals of R and F is a real-valued function of two variables defined on J × J ′ then for each pair A, B of self-adjoint operators on finite-dimensional complex inner product spaces X and Y respectively, with spectra contained in J and J ′ respectively, there is defined a selfadjoint operator F (A, B) on the tensor product space X ⊗ Y . One refers to the "operator function" F of two variables.There is a substantial literature concerning these operator functions and their properties. A function f : J → R is said to be operator monotone if f (A) ≤ f (B) whenever the terms are defined and A ≤ B. There is also a natural concept of operator convexity which for functions of one variable is intimately related to operator monotonicity. (Formal definitions are given in subsequent sections.) The present paper is concerned with the Fréchet differentiability and operator convexity of operator functions.In 1934 Löwner [13], in a celebrated paper, characterised those functions f : J → R which are operator monotone; they are, in particular, analytic. Several proofs of Löwner's central result are presented in a monograph by Donoghue [5]. More recently Hansen and Pedersen [9] have obtained yet another and very interesting proof and their development is followed in [4]. Part I of this paper is in part a response to their paper; it prompted the present authors to ask to what extent the results of the theory of operator monotone and operator convex functions can be obtained by exploiting the calculus. Theorems 2.1 and 4.2 include the results that, in both the one and two-variable situations, if a real-valued function is continuously L times differentiable then the associated operator functions are L times Fréchet differentiable with continuous Fréchet derivatives. These theorems fill a longstanding gap in the theory. They allow straightforward and direct uses of the calculus in contexts where in the past ad hoc substitutes for the calculus have often been used. In particular the elementary differential conditions for monotonicity and convexity of real-valued functions extend naturally to operator functions (Theorems 3.1 and 3.2). Theorems 4.2 and 3.2 are exploited in Part II of the paper. Matrix forms of these results, previously obtained in the two-variable case by relatively ad hoc methods, follow immediately. The Fréchet differentiability of operator functions on infinite-dimensional Hilbert spaces is the subject of a paper by Hansen and Pedersen [8] but the overlap with

Differentiating Matrix Functions

“…For the proof, we require the following lemma. The result is well-known for onevariable functions, and Brown and Vasudeva prove this two-variable analogue in [3]: Lemma 4.6 Let J 1 and J 2 be open intervals in R and let f ∈ C m (J 1 ×J 2 , R). Choose j, k ∈ N with k ≤ j ≤ m. Let x 1 , .…”

“…The differentiability of matrix functions defined from one-variable functions is discussed frequently in the literature (see [2], [4], [6]). The most comprehensive result is by Brown and Vasudeva in [3] , who prove that m-times continuously differentiable real functions induce m-times continuously Fréchet differentiable matrix functions.…”

Operator Inequalities Associated with Jensen’s Inequality

“…The above result were known for k = 2 as it can be derived from a slight generalization of [7,Theorem 3.2] as noticed in [14,Theorem 4.4] where the authors gave a different proof.…”

Higher order extension of Löwner’s theory: Operator 𝑘-tone functions

“…In the infinite-dimensional setting, the existence of the kth derivative is rather subtle but it exists in the operator norm when f is analytic (see [24] for the kth derivative under a weaker assumption). It is well known [9] that the operator/matrix monotonicity of f is characterized by the positivity of derivative (0.1) for k = 1, and the operator/matrix convexity is similarly characterized by the positivity of (0.1) for k = 2. Our motivation for the present paper came from the naive question of what is a higher order extension of the operator monotonicity related to the higher order derivative given in (0.1) for k > 2.…”

“…The above kth derivative exists for matrices whenever f is C k on (a, b) (see [9,18]). In the infinite-dimensional setting, the existence of the kth derivative is rather subtle but it exists in the operator norm when f is analytic (see [24] for the kth derivative under a weaker assumption).…”