Weak-star Topology Research Articles

Notion of quadratic variation in Banach spaces

This article introduces the notion of global quadratic variation which is based on a notion weaker than the usual tensor topology appearing in the literature of stochastic processes in Banach separable spaces. It is based on convergence in the weak-star topology of approximating sequences. This calculus pursues in the infinite dimensional space the stochastic calculus via regularization introduced by Russo and Vallois for real valued processes and developed in Banach space by Di Girolami and Russo. This article in particular focuses on the relation with classical calculus in order to compare our global quadratic variation with the classical existing notions of quadratic variations in the literature.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Ensemble approximations for constrained dynamical systems using Liouville equation

For a class of state-constrained dynamical systems described by evolution variational inequalities, we study the time evolution of a probability measure which describes the distribution of the state over a set. In contrast to smooth ordinary differential equations, where the evolution of this probability measure is described by the Liouville equations, the flow map associated with the nonsmooth differential inclusion is not necessarily invertible and one cannot directly derive a continuity equation to describe the evolution of the distribution of states. Instead, we consider Lipschitz approximation of our original nonsmooth system and construct a sequence of measures obtained from Liouville equations corresponding to these approximations. This sequence of measures converges in weak-star topology to the measure describing the evolution of the distribution of states for the original nonsmooth system. This allows us to approximate numerically the evolution of moments (up to some finite order) for our original nonsmooth system, using a solver that uses finite order moment approximations of the Liouville equation. Our approach is illustrated with the help of two academic examples.

A note on weak-star and norm Borel sets in the dual of the space of continuous functions

Let Bo ⁡ ( T , τ ) \operatorname {Bo}(T,\tau ) be the Borel σ \sigma -algebra generated by the topology τ \tau on T T . In this paper we show that if K K is a Hausdorff compact space, then every subset of K K is a Borel set if and only if \[ \operatorname {Bo}(C^*(K),\mathnormal {w}^*)=\operatorname {Bo}(C^*(K),\left \|\cdot \right \|),\] where \mathnormal {w}^* denotes the weak-star topology and ‖ ⋅ ‖ \left \|{\cdot }\right \| is the dual norm with respect to the sup-norm on the space of real-valued continuous functions C ( K ) C(K) . Furthermore, we study the topological properties of the Hausdorff compact spaces K K such that every subset is a Borel set. In particular, we show that if the axiom of choice holds true, then K K is scattered.

Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.

Arcwise connectedness of the set of ergodic measures of hereditary shifts

We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology, and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok.

On the convergence of soft potential dynamics to hard sphere dynamics

We address a question raised in the work of Gallagher, Saint-Raymond and Texier (From Newton to Boltzmann: Hard Spheres and Short-range Potentials, Zurich Lectures in Advanced Mathematics, EMS, 2013) that concerns the convergence of soft-potential dynamics to hard sphere dynamics. In the case of two particles, we establish that hard sphere dynamics is the limit of soft sphere dynamics in the weak-star topology of BV. We view our result as establishing a topological method by which to construct weak solutions to the ODE of hard sphere motion.

О свойстве плотности в пространствах слабо абсолютно непрерывных мер. Общий случай

It is shown that some set of all step functions (and the set of all uniform limits of such functions) allows an embedding into a compact subset (with respect to weak-star topology) of the set of all finitely additive measures of bounded variation in the form of an everywhere dense subset. In particular, we consider the set of all step functions (the set of all uniform limits of such functions) such that an integral of absolute value of the functions with respect to nonnegative finitely additive measure λ is equal to unity. For these sets, the possibility of embedding is proved without any additional assumptions on λ; this generalizes the previous results. Using the Sobczyk-Hammer decomposition theorem, we show that for λ with the finite range, the above-mentioned sets of functions allow an embedding into the unit sphere (in the strong norm-variation) of weakly absolutely continuous measures with respect to λ in the form of an everywhere dense subset. For λ with an infinite range, the above-mentioned sets of functions allow an embedding into the unit ball of weakly absolutely continuous measures with respect to λ in the form of an everywhere dense subset. The results can be helpful for an extension of linear impulse control problems in the class of finitely additive measures to obtain robust representations of reachable sets given by constraints of asymptotic character.

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball BX* of X* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.

Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

On the zeros of asymptotically extremal polynomial sequences in the plane

Let E be a compact set of positive logarithmic capacity in the complex plane and let {Pn(z)}1∞ be a sequence of asymptotically extremal monic polynomials for E in the sense that lim supn→∞‖Pn‖E1/n≤cap(E). The purpose of this note is to provide sufficient geometric conditions on E under which the (full) sequence of normalized counting measures of the zeros of {Pn} converges in the weak-star topology to the equilibrium measure on E, as n→∞. Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of E has a single component and the boundary of this component has an “inward corner” (more generally, a “non-convex singularity”). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences.

A note on monolithic scattered compacta

For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP, for short) is that the dual unit ball BE⁎ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X turns out to be sufficient for C0(X) to enjoy the CSCP.

Ordinary semicascades and their ergodic properties

A relationship is considered between ergodic properties of a discrete dynamical system on a compact metric space Ω and characteristics of companion algebro-topological objects, namely, the Ellis enveloping semigroup E, the Kohler enveloping operator semigroup Γ, and the semigroup G being the closure of the convex hull of Γ in the weak-star topology on the operator space EndC*(Ω). The main results are formulated for ordinary (having metrizable semigroup E) semicascades and for tame dynamical systems determined by the condition cardE ⩽ c. A classification of compact semicascades in terms of topological properties of the semigroups specified above is given.

A sequence of discrete minimal energy configurations that does not converge in the weak-star topology

We demonstrate a set A and a value of s for which any sequence of N-point discrete minimal Riesz s-energy configurations on A does not have an asymptotic distribution in the weak-star sense as N tends to infinity.

Zero Temperature Limits of Gibbs Equilibrium States for Countable Markov Shifts

We prove that, given a uniformly locally constant potential f on a countable state Markov shift and suitable conditions which guarantee the existence of the equilibrium states μtf for all t, the measures μtf converge in the weak star topology as t tends to infinity.

Problems with relaxed shifted controls and the pointwise maximum principle

In previous work [J. Warga, A proper relaxation of controls with variable shifts, J. Math Anal. Appl. 196 (1995) 783–793], it has been shown that controls with k variable and unrelated time shifts form a set ℜ whose closure ℜ ¯ in the weak star topology of L 1 ( T , C ( Ω k + 1 ) ) ∗ is a convex and compact space. This provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls. Similarly, the necessary optimality conditions, for original and relaxed minima, that define an extremal, previously studied for differential and functional equations, apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only. In the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal (ordinary or relaxed) shifted controls acting through ordinary differential and certain functional–differential equations.

Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

Let A be a compact set in R p of Hausdorff dimension d. For s ∈ ( 0 , d ) , the Riesz s-equilibrium measure μ s , A is the unique Borel probability measure with support in A that minimizes the double integral over the Riesz s-kernel | x − y | − s over all such probability measures. In this paper we show that if A is a strictly self-similar d-fractal, then μ s , A converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.

Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d

Let A be a compact set in \({\mathbb R}^{p}\) of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ s is the unique Borel probability measure with support in A that minimizes $$ {I_s}(\mu):=\int\int{\frac{1}{{|{x} - {y}|}^{s}}}d\mu(y)d\mu(x) $$ over all such probability measures. If A is strongly \(({\mathcal H}^d, d\kern.5pt)\)-rectifiable, then μ s converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.

Remarks on Littlewood–Paley Analysis

AbstractLittlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

Zeros of the derivatives of Faber polynomials associated with a universal covering map

For a compact set E ⊂ C with connected complement, we study asymptotic behavior of normalized zero counting measures { μ k } of the derivatives of Faber polynomials associated with E. For example if E has empty interior, we prove that { μ k } converges in the weak-star topology to a probability measure whose support is the boundary of g ( D ) , where g : { | z | > r } ∪ { ∞ } → C ¯ ⧹ E is a universal covering map such that g ( ∞ ) = ∞ and D is the Dirichlet domain associated with g and centered at ∞ . Our results are counterparts of those of Kuijlaars and Saff [Asymptotic distribution of the zeros of Faber polynomials, Math. Proc. Cambridge Philos. Soc. 118 (1995) 437–447] on zeros of Faber polynomials.