Matrix convexity of functions of two variables

Aujla, Jaspal Singh

doi:10.1016/0024-3795(93)90119-9

Cited by 14 publications

(12 citation statements)

References 6 publications

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“…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.…”

Section: K and A(i) T Denotes The Transpose Of A(i) Consequentmentioning

confidence: 82%

Operator Inequalities Associated with Jensen’s Inequality

Hansen

2000

Survey on Classical Inequalities

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Abstract. We give a survey of various operator inequalities associated with Jensen's inequality and study the class of operator convex functions of several variables. Related questions are considered. Jensen's classical inequalityLet I be a real interval. A function f : I → R is said to be convex, iffor all t, s ∈ I and every λ ∈ [0, 1]. Notice that the definition, in order to be meaningful, requires that f can be evaluated in λt + (1 − λ)s, or equivalently that I is convex. But this is satisfied because the convex subsets of R are the intervals. If f satisfies (1.1) just for λ = 1/2, then f is said to be midpoint convex. It is easy to establish that a continuous and mid-point convex function is convex. The geometric interpretation of (1.1) is that the graph of f is below the chord and consequently above the extensions of the chord. This entails that a convex function defined on an open interval is continuous. Condition (1.1) can be reformulated asfor all mutually different numbers t, s, r ∈ I. If we for such numbers define the divided difference [ts] of f taken in the points t, s as

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“…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.…”

Section: K and A(i) T Denotes The Transpose Of A(i) Consequentmentioning

confidence: 82%

Operator Inequalities Associated with Jensen’s Inequality

Hansen

2000

Survey on Classical Inequalities

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“…However, the verification that a function of two variables is operator convex may not be easy. The following interesting theorem was deduced, in a slightly more restricted form, by Aujla [2] from a more limited result obtained by Ando [1]. It will be shown that the calculus provides a natural and straightforward proof of the theorem, quite different from the original proof.…”

Section: Definition Let J and Jmentioning

confidence: 97%

“…It is shown that the calculus provides a natural proof of a theorem, deduced by Aujla [2] from a result of Ando [1], which identifies certain operator convex functions of two variables.…”

Section: Operator Functions Of Two Variablesmentioning

confidence: 99%

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The calculus of operator functions and operator convexity

Brown

Vasudeva

2000

Dissertationes Math.

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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

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“…It is also clear that the pointwise limit of operator convex functions of n variables is operator convex. Some operator convex functions of two variables have been studied by Ando [2] and Aujla [3] who also characterized the separately operator convex functions of two variables in terms of an operator inequality. In the case of one variable, it is well known that a continuous function f : [0, α[→ R is operator convex (and f (0) ≤ 0), if and only if it satisfies the inequality…”

Section: Functional Calculus For Functions Of Several Variablesmentioning

confidence: 99%