nwu-puk mathematics workshop: abstract · pdf filemokhwetha mabula { left-k-sequential...

NWU-PUK Mathematics Workshop:Abstract Analysis

18-21 September 2013

Unit for BMI, NWU

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Contents

General information 3Program 4Abstracts 5Jurie Conradie – Characterizations of uniform integrability 5Richard de Beer – Maximal Ergodic Inequalities for Banach function spaces 5Ben de Pagter – Translation invariant Banach function spaces 5Joe Diestel – Some results in Descriptive Set Theory, as applied to Banach

spaces 5Jan Fourie – On almost p-convergent operators on Banach lattices 5Koos Grobler – Stochastic processes in Riesz spaces: The Kolmogorov-

Centsov theorem and Brownian motion 6Eder Kikianty – Order Generalised Gradient 6Coenraad Labuschagne – Fremlin’s tensor product of vector lattices 7Louis Labuschagne – A crossed product approach to Orlicz spaces 7Lenore Lindeboom – The role of the exponential function in generalised and

σg-Drazin inverses 7Mokhwetha Mabula – Left-K-sequential completeness and precompactness

in finite dimensional asymmetrically normed lattices 7Charles Maepa – On character and approximate character amenability of

various Segal algebras 8W ladys law Adam Majewski – On applications of Orlicz Spaces to Statistical

Physics 8Miek Messerschmidt – A stronger Open Mapping Theorem with applications

in ordered Banach spaces 8Hermann Rabe – Asymptotics of the smallest singular value of a class of

Toeplitz-generated matrices 9Heinrich Raubenheimer – Positive Riesz Operators 9Sanne ter Horst – Pre-order and equivalence relation invariance of linear

fractional Redheffer maps 10Bruce Watson – Martingales and their generalizations in the context of

Riesz spaces 10Quanhua Xu – A noncommutative Helson-Szego theorem with applications

to Toeplitz operators 10Campus map 11Potchefstroom map 12

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General information

All lectures will take place in venue G1-204 (see the included campus map). Teaand coffee will be served in the tea-room on level 2 of building G1 during the coffeebreaks. The workshop dinner (in the form of a spit-roast sheep) will be held atbuilding F-22. A cash bar will be available at that venue for the duration of thedinner. Participants are encouraged to use the lunch break for discussions, andfor this reason there is no official lunch venue. Rather we encourage participantsto link up with one another and avail themselves of the many restaurants in theimmediate vicinity of the campus. There are many restaurants and fast food outletsclustered around the area marked with a D on the campus map. In addition thereis a student cafeteria as well as a coffee bar housed in cluster of buildings markedF-14.

Day 1 18 September 2013

9:00-9:20 Registration

9:20-9:40 Opening

9:45-10:40 Plenary 1.1 De Pagter

10:45-11:15 Tea

11:15-12:10 Plenary 1.2 Grobler

12:15-12:55 Lecture 1.1 Messerschmidt

13:00-15:15 Lunch

15:15-15:55 Lecture 1.2 Kikianty

16:00-16:30 Tea

16:35-17:15 Lecture 1.3 Ter Horst


9:00-9:55 Plenary 2.1 Raubenheimer

10:00-10:40 Lecture 2.1 Mabula

10:45-11:15 Tea

11:15-12:10 Plenary 2.2 Watson

12:15-12:55 Lecture 2.2 Conradie

13:00-15:30 Lunch

15:30-16:00 Tea

16:00-17:30 Problem Session

18:00-23:00 Dinner


9:00-9:55 Plenary 3.1 C. Labuschagne

10:00-10:40 Lecture 3.1 L. Labuschagne

10:45-11:15 Tea

11:15-12:10 Plenary 3.2 Xu

12:15-12:55 Lecture 3.2 Rabe

13:00-15:15 Lunch

15:15-15:55 Lecture 3.3 De Beer

16:00-16:30 Tea

16:35-17:15 Lecture 3.4 Maepa


9:00-9:55 Plenary 4.1 Majewski

10:00-10:40 Lecture 4.1 Lindeboom

10:45-11:15 Tea

11:15-12:10 Plenary 4.2 Diestel

12:15-12:55 Lecture 4.2 Fourie

NWU-PUK Workshop: Program

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ABSTRACTS

Characterizations of uniform integrability

Jurie Conradie

University of Cape Town, Rondebosch, South Africae-mail: [email protected]

A theorem of de la Vallee Poussin characterizes a uniformly integrable subsetof L1 ∗ (X,Σ, µ), with (X,Σ, µ) a probability space, as a bounded subset of an

Orlicz space LΦ(X,Σ, µ), where limt→∞Φ(t)t = ∞. We investigate the possibility

of replacing this class of Orlicz spaces by other classes of Banach function spaces.The fact that uniform integrability can be characterized in terms of rearrangementsof functions is crucial to the generalization, and allows us to deduce similar resultsin the setting of non-commutative function spaces.

Maximal Ergodic Inequalities for Banach function spaces

Richard de Beer

Department of Decision Sciences, UNISAe-mail: [email protected]

We analyse the Transfer Principle, which is used to generate weak type maximalinequalities for ergodic operators, and extend it to the general case of σ compactlocally compact groups acting measure preservingly on σ finite measure spaces.We show how the techniques developed here generate various weak type maximalinequalities on different Banach function spaces, and how the properties of thesefunction spaces influence the weak type inequalities that can be obtained. Finally,we demonstrate how the techniques developed imply almost sure pointwise conver-gence of a wide class of ergodic averages.

Translation invariant Banach function spaces

Ben de Pagter

University of Technology Delftemail: [email protected]

In this talk we will discuss some properties of so-called translation invariantBanach function spaces on compact abelian groups. This class of Banach functionspaces includes the rearrangement invariant spaces on such groups, but is muchlarger and exhibits different features. Translation invariant Banach function spacesarise naturally in the study of Fourier multiplier operators.

The talk is based on joint work with Werner Ricker.

Some results in Descriptive Set Theory, as applied to Banach spaces

Joe Diestel

Kent State University, Ohioe-mail: j [email protected]

Some results in Descriptive Set Theory, as applied to Banach spaces Abstract: Iwill discuss some notions of decriptive set theory that can be applied to problemsin Banach space theory. In particular, I will talk about applying some ideas tothe Radon Nikodym Property and related notions. In addition to recalling somestriking stability results, I hope to describe certain non universality results. Muchof this work is classical; that which is not is due to B. M. Braga and is quite new.

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On almost p-convergent operators on Banach lattices

Jan Fourie

NWU-Potchefstroome-mail: [email protected]

The concept “p-convergent operator” on Banach spaces (for 1 ≤ p < ∞) wasintroduced by J. Castillo and F. Sanchez in a paper “Dunford-Pettis-like Propertiesof Continuous Vector Function Spaces” in Revista Matematica de la UniversidadComplutense de Madrid 6(1) (1993). In recent joint work with Elroy Zeekoei,we observed that each bounded linear operator from a Banach space X to thespace c0 is p-convergent if and only if the scalar sequence (x∗n(xn))n converges to0 for all weak∗ null sequences (x∗n) in X∗ and all weakly-p-summable sequences(xn) in X and equivalently, that this is the case when every symmetric bilinearseparately compact map X × X → c0 is p-convergent. Banach spaces for whichthis is true, are said to have the ∗-Dunford Pettis property of order p (or briefly,they have the DP ∗Pp). In this talk we introduce weak p-convergent and almost p-convergent operators on Banach lattices and discuss the relationship between them.Characterizations of Banach spaces for which weakly p-summable sequences arenorm null sequences are well known in the literature. We characterize the Banachlattices for which all positive disjoint weakly p-summable sequences are norm nullsequences, resulting in a discussion of the so called positive Schur property of orderp. Almost p-convergent operators will play important role in this discussion.

Stochastic processes in Riesz spaces: The Kolmogorov-Centsovtheorem and Brownian motion

Koos Grobler


We consider continuous time stochastic processes on a Dedekind complete Rieszspace on which a conditional expectation is defined. The theorem mentioned in thetitle is used in the classical theory to prove the existence of a Brownian motion.We prove an abstract version of this theorem and show how this can be used toextend the notion of Brownian motion as defined by Labuschagne and Watson fordiscrete processes to continuous time stochastic processes.

Order Generalised Gradient

Eder Kikianty

School of Computational and Applied Mathematics, WITSe-mail: [email protected]

The subgradient inequality is expressed in terms of the subgradient vector, adescription for the one-sided directional derivative, and is used to characterises theconvexity of a function. In this talk, I will present the notion of order generalisedgradient for operator-valued functions, which is a generalisation of the subgradientinequality in the settings of bounded self-adjoint operators. We state the connectionbetween the order generalised gradient and the Gateaux derivative of operator-valued functions. Evidently, we state a characterisation of convexity analogues tothe subgradient inequality in the context of operator-valued functions.

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Fremlin’s tensor product of vector lattices

Coenraad Labuschagne

School of Computational and Applied Mathematics, WITSe-mail: [email protected]

We will present an overview of Fremlin’s tensor product of Archimedean vectorlattices. Properties of Fremlin’s positive projective norm on the tensor product ofBanach lattices will be included.

A crossed product approach to Orlicz spaces

Louis Labuschagne


We show how the known theory of noncommutative Orlicz spaces for semifinitevon Neumann algebras equipped with an fns trace, may be recovered using crossedproduct techniques. Then using this as a template, we construct analogues of suchspaces for type III algebras. The constructed spaces naturally dovetail with andclosely mimic the behaviour of Haagerup Lp-spaces.

The role of the exponential function in generalised and σg-Drazininverses

Lenore Lindeboom

Department of Mathematical Sciences, UNISAe-mail: [email protected]

Using the definition of the exponential function of an element a in a Banachalgebra A denoted by exp a and the fact that

max{<λ∣∣∣∣λ ∈ σ(a)} = inf

t>0

1

tlog ‖exp ta‖ = lim

t→∞

1

tlog ‖exp ta‖

we obtain a specific equation for the g −Drazin inverse aD for each ofthe cases limt→∞ exp ta = 0 and limt→∞ exp ta = p.Then in the case where σ is an isolated spectral set of a such that0 ∈ Res(a) ∪ σ together with Res{σ(a)\σ} < 0 we obtain a similar equation.

Reference:A.Dajic and J J Koliha, The σg-Drazin inverse and the generalized Mbekhta

decomposition, Integral Equations and Operator Theory 57(2007), 309-326.

Left-K-sequential completeness and precompactness in finitedimensional asymmetrically normed lattices

M. Mabula

Department of Mathematics and Applied Mathematics, University of Pretoriae-mail: [email protected]

If (X, ‖.‖) is a real normed lattice, then p(x) = ‖x+‖, where x ∈ X and x+ = x∨0defines an asymmetric norm on X. We call the pair (X, p) an asymmetricallynormed lattice. The topology τp on X induced by the quasi-metric dp defined bydp(x, y) = p(y − x), x, y ∈ X, is never T1 and therefore the limits of convergencesequences are not unique.

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We study precompactness and left-K-sequential completeness in finite dimen-sional asymmetrically normed lattices. We characterize precompact subsets of Rm

and obtain a sufficient condition for bounded above left-K-sequential complete sub-sets of Rm.

On character and approximate character amenability of various Segalalgebras

O.T. Mewomo

Department of Mathematics, Federal University of Agriculture Abeokuta, Nigeriae-mail: [email protected]

S.M. Maepa ∗

Department of Mathematics and Applied Mathematics, University of Pretoriae-mail: [email protected]

M.O. Uwala

Department of Mathematics, Federal University of Agriculture Abeokuta, Nigeriae-mail: [email protected]

We investigate character and approximate character amenability of various Segalalgebras in both the group algebra L1(G) and the Fourier algebra A(G) of a locallycompact group G.

On applications of Orlicz Spaces to Statistical Physics

W ladys law Adam Majewski

Institute of Theoretical Physics and Astrophysics, University of Gdanske-mail: [email protected]

(joint work with L.E. Labuschagne)

A new rigorous approach based on Orlicz spaces for the description of the sta-tistics of large regular statistical systems, both classical and quantum, will be pre-sented. The Orlicz spaces (commutative as well as noncommutative) we explicitlyuse are built on exponential function (for description of regular observables) and onentropic type function (for the corresponding states). They form a dual pair (bothfor classical and quantum systems). The relation to Boltzmann theory is indicated.Moreover, it is shown that the latter Orlicz space selects states with well definedentropy function. The key tools are Banach function spaces and noncommutativeintegration theory.

A stronger Open Mapping Theorem with applications in orderedBanach spaces

Miek Messerschmidt

Leiden Universitye-mail: [email protected]

The usual Banach–Schauder Open Mapping Theorem can be formulated as fol-lows:

Theorem Let X and Y be Banach spaces and T : Y → X a bounded linear map.Then the following are equivalent:

(1) T is surjective;(2) T is open.

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We will indicate how this can be combined with Michael’s Selection Theorem toyield the following stronger and more general Open Mapping Theorem:

Theorem Let X and Y be Banach spaces and let C ⊂ Y be a closed (not necessarilyproper) cone. Let T : C → X be a continuous additive positively homogeneous map.Then the following are equivalent:

(1) T is surjective;(2) T is open;(3) There exist K > 0 and a continuous positively homogeneous map γ : X →

C, such that T ◦ γ(x) = x and ‖γ(x)‖ ≤ K‖x‖, for all x ∈ X.

This readily implies the following:

Theorem Let X be a Banach space and {Ci}i∈I be an arbitrary collection of closedcones in X, such that every x ∈ X can be written as an absolutely convergent seriesx =

∑i∈I ci, where ci ∈ I, for all i ∈ I. Then there exist a constant K > 0 and

continuous positively homogeneous maps γi : X → Ci (i ∈ I), such that:

(1) x =∑

i∈I γi(x), for all x ∈ X;(2)

∑i∈I ‖γi(x)‖ ≤ K‖x‖, for all x ∈ X.

If X is an ordered Banach space with a closed (not necessarily proper) generatingcone C, then Ando’s Theorem [1, Lemma 1] asserts that there exists K > 0 withthe property that, for all x ∈ X, there exist x± ∈ C such that x = x+ − x− and‖x±‖ ≤ K‖x‖. As a consequence of the above result, there exists such a boundeddecomposition that is, in addition, continuous and positively homogeneous.

This is joint work with Marcel de Jeu.

[1] T. Ando, On fundamental properties of a Banach space with a cone. PacificJ. Math. 12:1163-1169, (1962).

Asymptotics of the smallest singular value of a class ofToeplitz-generated matrices

Hermann Rabe


Square matrices of the form Xn = Tn +fn(T−1n )∗, where Tn is an n×n invertible

banded Toeplitz matrix and fn some positive sequence are considered. Convergencevia an order estimate is proven for the difference of ‖X−1

n ‖ and a function dependantonly on fn. Fredholmness of the infinite counterpart of Tn is shown to greatly affectthis result.

Positive Riesz Operators

Heinrich Raubenheimer

University of Johannesburge-mail: [email protected]

Let E be a Banach lattice and suppose S and T are operators defined on E suchthat 0 ≤ S ≤ T . We discuss conditions under which S is a Riesz operator if T is aRiesz operator.

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Pre-order and equivalence relation invariance of linear fractionalRedheffer maps

S. ter Horst


In a 1980 paper, Yu.L. Shmul’yan introduced a pre-order relation on the setof Hilbert space contractions, and proved that linear fractional transformationsof Redheffer type preserve this pre-order. Motivated by this paper, a pre-orderrelation is introduced on the set of Schur class functions, that is, analytic functionson the unit disc of C whose values are contractive Hilbert space operators, andthe behavior of Redheffer transformations with respect to this pre-order and theassociated equivalence relation is studied.

Martingales and their generalizations in the context of Riesz spaces

Bruce Watson

School of Mathematics, WITSe-mail: [email protected]

In this talk we discuss martingales, sub and super martingales, asymptotics mar-tingales and mixingales in both the classical and Riesz space contexts. How theseobjects relate to each other will be discussed as well as the related decompositionand convergence results.

A noncommutative Helson-Szego theorem with applications to Toeplitzoperators

Quanhua Xu

Universite de Franche-Comte and Wuhan Universityemail: [email protected]

We present a noncommutative version of the classical Helson-Szego theoremabout the angle between past and future for subdiagonal subalgebras. We thenproceed to use this theorem to characterize the symbols of invertible Toeplitz oper-ators on the noncommutative Hardy spaces associated to subdiagonal subalgebras.

nwu-puk mathematics workshop: abstract · pdf filemokhwetha mabula { left-k-sequential...

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