Positive Operators, Riesz Spaces and Applications

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Topics in Functional Analysis & Positive Operators, Riesz Spaces and Applications


Masamu Program: Topics in Functional Analysis - Virtual Course - Fall 2020

Syllabus for the course in Topics in Functional Analysis to be given at University of Pretoria, South Africa, August 5, 2020 to December 16, 2020. This course is sponsored by the Masamu Program: US-Africa Collaborative Research Network

Course will meet every Wednesday at 17:00 -19:00 Hours (5pm-7pm) South African time.
Course will start: Wednesday, August 5, 2020
Course will end:  Wednesday, December 16, 2020

Lectures for this course will be given by:

Geraldo Soares de Souza (PhD)
Professor Emeritus
Department of Mathematics
Auburn University
Auburn, AL USA 36849
Email: [email protected]

Eddy Kwessi (PhD)
Associate Professor
Department of Mathematics
Trinity University
San Antonio, TX USA 78212
Email: [email protected]

Contents:

1 . Vector Spaces
2.  Metric and Normed Spaces
3.  Banach and Hilbert Spaces
4.  Sequences Spaces and   compactness on sequence spaces 
5.  Linear Transformations and characterization of bounded linear functionals
6.  Hahn Banach Theorem
7.  Closed Graph Theorem
8.  Open Mapping Theorem
9.  Weak Topologies 
10.  Uniform Boundedness Principle
11.  Equivalence of Some Banach Spaces
12. Exercises

Note:

This course will be based on the lectures of “Mini-Course on Functional Analysis” given by Prof. Geraldo de Souza at the SAMSA Masamu Advanced Study Institute held in Livingstone, Zambia, December 2-12, 2011. Course notes will be available to all students.

South Africa Local Coordinator:

Mokhwetha Mabula (PhD)
Senior Lecturer of Mathematics
University of Pretoria
Pretoria, South Africa
Email: [email protected]  

Masamu Coordinator:

Overtoun Jenda (PhD)
Assistant Provost for Special Projects and Initiatives and Professor of Mathematics
Auburn University
Auburn, AL USA 36849
Email: [email protected]

Note from Professor Overtoun Jenda:

This course is sponsored by the SAMSA Masamu Program (https://www.masamu.auburn.edu/). Participants successfully completing the course will receive a certificate of participation and will be eligible to attend the Masamu Advanced Study Institute (MASI) and Workshops series.


Positive Operators, Riesz Spaces and Applications

A workshop took place, from 30 June to 3 July 2021, to establish a group of researchers in the fields of Positive operators, Riesz Spaces (or more general ordered vector spaces) and their applications in Economics and Finance between the University of the Free State, University of Fort Hare, University of Johannesburg, University of Limpopo, Sefako Makgatho University and University of Pretoria. To recruit and train graduates students and also take part in the Annual MASAMU research network in collaborations with Auburn University in USA and Southern African Mathematical Science Association (SAMSA).

BACKGROUND

The theory of positivity in modern analysis was developed in the 70s of the 20th century, mainly by Zaanen with his collaborators, but also by Russian and Poland schools. This resulted in the development of Riesz space (vector lattice) theory, and soon thereafter, the applications of positivity theory were found in the spectral theory of positive operators and economics. In 1990, Aliprantis and his team of collaborators held a mini conference on positive operators, Riesz Spaces and their applications, inviting  Riesz space theorists and Economists working on equilibrium problems to exchange ideas and share experience .  Thereafter, many discoveries were made resulting in the publications of several articles and books in these areas.  Some seminal books on Riesz Spaces with applications to Economics were written by Aliprantis and his collaborators, notably the extended book on Locally Solid Riesz Spaces and Applications to Economics (2003). In 2000s, applications of Riesz Spaces to stochastic processes were developed mainly by C.C.A. Labuschagne and his team of researchers, as well as by Troitsky and others.  Labuschagne and his PhD students developed an idea in stochastic processes in Riesz spaces that provided new and more information about the martingale convergence in Bochner spaces. Recently, applications of Riesz Spaces in asymmetrically normed lattices were given and some progress has been achieved in this direction, in particular concerning the notion of compactness in such spaces. The main motivation in the study of these comes from their applications to theoretical computer science. The major reference on recent developments in this area is the book Functional Analysis in Asymmetric Normed Spaces (2013) by S. Cobzas.

AIMS

Our aim is to establish a group of researchers in the fields of Positive operators, Riesz Spaces (or more general ordered vector spaces) and their applications in Economics and Finance between the University of the Free State, University of Fort Hare, University of Johannesburg, University of Limpopo, Sefako Makgatho University and University of Pretoria. To recruit and train graduates students and also take part in the Annual MASAMU research network in collaborations with Auburn University in USA and Southern African Mathematical Science Association (SAMSA).

PROJECTS/PROBLEMS

The main focus is on development and extension of results in Riesz Spaces to more general ordered vector spaces,

  1. The study of bands in ordered vector spaces which are not necessarily Riesz Spaces.
  2. The study of cones, bases of cones and their applications in characterization of Reflexivity and equilibrium problems in economics.
  3. Development of measure free setting Martingales in Riesz spaces.
  4. Asymmetric norm theory on Normed Riesz Spaces.
  5. Study of utility functions in ordered vector spaces.
  6. Fixed point theory on Riesz Spaces (or an ordered vector spaces) and on asymmetric setting.
  7. Applications of Riesz spaces in spectral theory of positive operators.

WORK PLAN/ACTIVITIES

The following activities will be followed,

  1. Organize at least one mini workshop in a year to be held at University of Pretoria, Hatfield campus.
  2. Face to face research meeting/discussion/seminar amongst the members within traveling distance.
  3. Skype or online research meeting with members whose Universities is distant from the rest of the group.
  4. Participate in Positivity conference, Topology, Algebra, Analysis and Geometry (TAAG) conference, Southern African Mathematical Science Association (SAMSA) conference, South African Mathematical Society, Topology and its application conference.

COLLABORATIONS/NETWORKS

The departments of Mathematics, Computer Science and Economics at the following Universities are our partners

  1. Babes Bolyai University, Cluj, Romania
  2. Public University of Navarre, Pamplona, Spain
  3. Balearic Islands University, Palma de Mallorca, Spain
  4. University of Valencia, Valencia, Spain
  5. Auburn University, USA.

Plenary Speakers/Lecturers

  1. Dr Abdullah Aydin, The Department of Mathematics, Mus Alparslan University, Mus, Turkey, a) Multiplicative Convergence on Riesz Spaces
    b) Order Convergence in f-algebras.
  2. Prof Nazife Erkusun,  Department of Mathematics, Hacettepe University, Ankara, Turkey
    a) On Ergodic properties of operator nets on Von Neumann Algebras.
    b) Unbounded p-convergence in normed Riesz Spaces.
  3. Prof Jacek Banasiak, Department of Mathematic s and Applied Mathematics, Univerisity of Pretoria.
    a) Banach lattices and Applications.
  4. Dr Mokhwetha Mabula,  Department of Mathematics and Applied Mathematics, University of Pretoria.
    a) Order convergence in Riesz Spaces and order continuous operators.
  5. Miss Queen Mabe, Department of Economics, University of Johannesburg
    a) Riesz spaces and Equilibrium equations.

Scientific Committee

  1. Prof Stefan Cobzas, Department of Mathematics and Computer Science, Babes Bolyai University, Romania
  2. Prof Geraldo De Souza, Department of Mathematics and Statistics, Auburn University, United States of America.
  3. Prof Oscar Valero, Department of Mathematics and Computer Science, Balearic Islands University, Spain
  4. Prof Enrique Sanchez-Perez, Department of Mathematics and Applied Mathematics,  Universitat Politecnica de Valencia, Spain
  5. Prof Esteban Indurain, Department of Statistics, Informatics and Mathematics, Public University of Navare, Spain
Published by Annel Smit

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