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.     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

Majorization and some operator monotone functions

Let h(t) be a non-decreasing function on I and k(t) an increasing function on J. Then h is said to be majorized by k if k(A)≦k(B) implies h(A)≦h(B). f(t) is operator monotone, by definition, if f(t) is majorized by t. By making use of this majorization we will show that is operator monotone on [0,∞) for 0≦a,b<∞ and for 0≦r≦1; the special case of a=b=1 is the theorem due to Petz-Hasegawa.     

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.     

On a correspondence between probabilistic and fuzzy choice functions

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to be sufficient to ensure that the associated fuzzy choice function is rational.     

Norm inequalities related to operator monotone functions

Let A,B be positive semidefinite matrices and any unitarily invariant norm on the space of matrices. We show for any non-negative operator monotone function f(t) on , and for non-negative increasing function g(t) on with g(0) = 0 and , whose inverse function is operator monotone. Received: 1 February 1999     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

Jensen’s inequality for operator monotone functions

In this paper, we shall show that some classical inequalities for monotone functions also hold for operator monotone functions on an arbitrary Hilbert spaceH. Such results can be found in the classical book [1, p. 83] or in a new book [2].     

Duality between quasi-concave functions and monotone linkage functions

A function F defined on the family of all subsets of a finite ground set E is quasi-concave, if F(X∪Y)≥min{F(X),F(Y)} for all X,Y⊆E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, graph theory, data mining, clustering and other fields. The maximization of a quasi-concave function takes, in general, exponential time. However, if a quasi-concave function is defined by an associated monotone linkage function, then it can be optimized by a greedy type algorithm in polynomial time. Recently, quasi-concave functions defined as minimum values of monotone linkage functions were considered on antimatroids, where the correspondence between quasi-concave and bottleneck functions was shown Kempner and Levit (2003) [6]. The goal of this paper is to analyze quasi-concave functions on different families of sets and to investigate their relationships with monotone linkage functions.     

On generalized absolutely monotone functions

Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed. The research of the second author was partially supported by U.S. Army Contract-DA-31-124-ARO-D-462 in the MRC, Madison, Wisconsin.     

Partial inverse of a monotone operator

ForT a maximal monotone operator on a Hilbert spaceH andA a closed subspace ofH, the “partial inverse”T _A ofT with respect toA is introduced.T _A is maximal monotone. The proximal point algorithm, as it applies toT _A , results in a simple procedure, the “method of partial inverses”, for solving problems in which the object is to findx ∈ A andy ∈ A ^⊥ such thaty ∈ T(x). This method specializes to give new algorithms for solving numerous optimization and equilibrium problems.     

On monotone functions of tree structures

Let T be a rooted tree structure with n nodes a₁,…,a_n. A function f: {a₁,…,a_n} into {1 < ? < k} is called monotone if whenever a_i is a son of a_j, then f(a_i) ≥ f(a_j). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ?^?nn^{?$\text{3}\text{2}$} (n → ∞) are obtained for several classes of tree structures.