On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function f: R⁺ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra.     

Maximum Lebesgue extension of monotone convex functions

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a “nice” dual representation of the function.     

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.

     

On the Lebesgue property of monotone convex functions

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.     

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing $n\times n$ matrix functions $D(·)$ defined on some open interval $?=(a\text{,}b)$ of $\mathbb{R}$ and their limit values $D(a)$ and $D(b)$ at the endpoints a and b which are, in general, selfadjoint relations in ${\mathbb{C}}^n$. Certain space decompositions induced by the matrix function $D(·)$ are made explicit by means of the limit values $D(a)$ and $D(b)$. They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.     

A class of matrix monotone functions

In this paper we show that a class of functions is matrix (operator) monotone and give some consequences of it.     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

Differential analysis of matrix convex functions

We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1.     

On matrix measures and convex Liapunov functions

In this paper, we extend the concept of the measure of a matrix to encompass a measure induced by an arbitrary convex positive definite function. It is shown that this “modified” matrix measure has most of the properties of the usual matrix measure, and that many of the known applications of the usual matrix measure can therefore be carried over to the modified matrix measure. These applications include deriving conditions for a mapping to be a diffeomorphism on Rⁿ, and estimating the solution errors that result when a nonlinear network is approximated by a piecewise linear network. We also develop a connection between matrix measures and Liapunov functions. Specifically, we show that if V is a convex positive definite function and A is a Hurwitz matrix, then μ_V(A) < 0, if and only if V is a Liapunov function for the system $\text{x}\text{?}\text{= Ax}$. This linking up between matrix measures and Liapunov functions leads to some results on the existence of a “common” matrix measure μ_V(·) such that μ_V(A_i) < 0 for each of a given set of matrices A₁,…, A_m. Finally, we also give some results for matrices with nonnegative off-diagonal terms.     

The curse of dimensionality for the class of monotone functions and for the class of convex functions

We study the integration and approximation problems for monotone or convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in $d$) function values to achieve an error $\epsilon$.     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

On the structure of convex piecewise quadratic functions

Convex piecewise quadratic functions (CPQF) play an important role in mathematical programming, and yet their structure has not been fully studied. In this paper, these functions are categorized into difference-definite and difference-indefinite types. We show that, for either type, the expressions of a CPQF on neighboring polyhedra in its domain can differ only by a quadratic function related to the common boundary of the polyhedra. Specifically, we prove that the monitoring function in extended linear-quadratic programming is difference-definite. We then study the case where the domain of the difference-definite CPQF is a union of boxes, which arises in many applications. We prove that any such function must be a sum of a convex quadratic function and a separable CPQF. Hence, their minimization problems can be reformulated as monotropic piecewise quadratic programs.This research was supported by Grant DDM-87-21709 of the National Science Foundation.     

Remarks on convex functions and separable sequences,II

In this paper we generalize results on convex sequences and convex functions due to Wu and Debnath [S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl. 54 (2007) 525–534] by using some results on similarly separable sequences established in Niezgoda [M. Niezgoda, Remarks on convex functions and separable sequences, Discrete Math. 308 (2008) 1765–1773].     

A representation of maximal monotone operators by closed convex functions and its impact on calculus rules

We introduce a new representation for maximal monotone operators. We relate it to previous representations given by Krauss, Fitzpatrick and Mart??nez-Legaz and Théra. We show its usefulness for the study of compositions and sums of maximal monotone operators. To cite this article: J.-P. Penot, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,