Subdifferentials of Distance Functions, Approximations and Enlargements

In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set.     

Representation of generalized monotone multimaps

Abstract

We survey the role of generalized dualities when dealing with generalized monotone operators, observing that for many conjugacies the coupling function is neither bilinear nor finitely valued. We also make a comparison with the use of bifunctions considered in a similar perspective. We introduce a class of operators close to the class of accretive operators and we raise some open questions.     

Weaker Constraint Qualifications in Maximal Monotonicity

We give a sufficient condition, weaker than the others known so far, that guarantees that the sum of two maximal monotone operators on a reflexive Banach space is maximal monotone. Then we give a weak constraint qualification assuring the Brézis–Haraux-type approximation of the range of the sum of the subdifferentials of two proper convex lower-semicontinuous functions in nonreflexive Banach spaces, extending and correcting an earlier result due to Riahi.     

Bregman distances and Klee sets

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then–analogously to the Euclidean distance case–every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement the work by Hiriart-Urruty on the Euclidean case.     

Maurey-Rosenthal factorization of positive operators and convexity

We show that a Banach lattice X is r-convex, 1<r<∞, if and only if all positive operators T on X with values in some r-concave Köthe function spaces F(ν) (over measure spaces (Ω,ν)) factorize strongly through L_r(ν) (i.e., T=M_gR, where R is an operator from X to L_r(ν) and M_g a multiplication operator on L_r(ν) with values in F). This characterization of r-convexity motivates a Maurey-Rosenthal type factorization theory for positive operators acting between vector valued Köthe function spaces.     

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space is determined by the linear structure of .

     

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the nonsymmetric case.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

Weakly prime sets for function spaces

We define and study weakly prime sets for a function space and show that it coincides with the known concept of weakly prime sets for function algebras and spaces of affine functions.     

A sum theorem for (FPV) operators and normal cones

In his 2008 book “From Hahn-Banach to Monotonicity”, S. Simons mentions that the proof of Lemma 41.3, which was presented in the previous edition of his book “Minimax and Monotonicity” in 1998, is incorrect, and one does not know if this lemma and its consequences are true. The aim of this short note is not only to give a proof to the mentioned lemma but also to improve upon it by relaxing one of its assumptions.     

Extension of positive definite functions

Let Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ be an open, connected subset of Rⁿ${\mathbb{R}}^n$, and let F:Ω−Ω→C$F:\Omega -\Omega \to \mathbb{C}$, where Ω−Ω={x−y:x,y∈Ω}$\Omega -\Omega =\{x-y:x,y\in \Omega \}$, be a continuous positive definite function. We give necessary and sufficient conditions for F   to have an extension to a continuous positive definite function defined on the entire Euclidean space Rⁿ${\mathbb{R}}^n$. The conditions are formulated in terms of existence of a unitary representations of Rⁿ${\mathbb{R}}^n$ whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.     

Some results on topical functions and upward sets

The purpose of this paper is to introduce and discuss the concept of topical functions on upward sets.We give characterizations of topical functions in terms of upward sets.     

Bit-size estimates for triangular sets in positive dimension

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm.This extends the results by the first and the last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and interpolation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process.     

A Leibniz differentiation formula for positive operators

Is is shown that for n→+∞ the Leibnizian combination L′_n(fg)−fL′_n(g)−gL′_n(f) converges uniformly to zero on a compact interval W if the positive operators L_n belong to a certain class (including Bernstein, Gauss-Weierstrass and many others), and if the moduli of continuity of f,g satisfy ω_W(f;h)ω_W(g;h)=o(h) as h→0+. A counterexample shows that Lipschitz conditions are not appropriate to bring about a second-order version of this formula.     

Universality of holomorphic functions bounded on closed sets

In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.     

A characterization of maximum norms of commutators of positive contractions

In this note, we give a characterization of a pair (A,B) of positive contractions with commutator AB−BA of maximum possible norm. A necessary and sufficient condition is that either or its complex conjugate is in the closure of the numerical range of AB.     

Representation theorem for local operators in the space of continuous and monotone functions

We prove that every locally defined operator mapping the space of nondecreasing continuous functions into itself is a Nemytskii composition operator.