IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

We invite the Functional Analysis, Operator Theory and its Applications research community to propose Special Sessions. To propose a Special Session you need to submit, through the conference mail address, a proposal providing the following information:

- Title and brief description of the session,
- Names and e-mail addresses of the session organizers,
- Provisional list of speakers and titles of lectures (a typical number could be 10 speakers),
- Tentative number of the session participants.

The deadline for sessions submissions is **March 31st 2019.**

- Analysis and Algebraic Geometry for Operator Variables
*Igor Klep*, University of Auckland, New Zealand*Victor Vinnikov*, Ben-Gurion University, Beer-Sheva, Israel- Analysis and Synthesis for Operator Algebras
*Adam Dor-On*, University of Illinois at Urbana-Champaign, USA*Ruy Exel*, Universidade Federal de Santa Catarina, Brasil*Elias Katsoulis*, East Carolina University, USA*David Pitts*, University of Nebraska, Lincoln, USA.- Free Analysis and Free Probability
*Serban Belinschi*, Université Paul Sabatier, France*Mihai Popa*, University of Texas at San Antonio, USA*Roland Speicher*, Saarland University, Germany- Hypercomplex Analysis and Operator Theory
*Daniel Alpay*, Chapman University, Orange, CA, USA*Fabrizio Colombo*, Politecnico di Milano, Italy*Uwe Kähler*, Universidade de Aveiro, Portugal- Linear Operators and Function Spaces
*Cristina Câmara*, Instituto Superior Técnico, Portugal*Marek Ptak*, University of Agriculture in Krakow, Krakow, Poland- Multivariable Operator Theory
*Joseph Ball*, Virginia Tech University, USA*Vladimir Bolotnikov*, College of William & Mary, Williamsburg, Virginia, USA- Operators of Harmonic Analysis, Related Function Spaces and Applications

Dedicated to Lars-Erik Persson on his 75th Birthday *Humberto Rafeiro*, United Arab Emirates University, UAE*Natasha Samko*, UiT The Arctic University of Norway- Operators on Reproducing Kernel Hilbert Spaces
*Nikolai Vasilevski*, CINVESTAV, Mexico*Kehe Zhu*, State University of New York at Albany, USA- structural properties of reproducing kernel Hilbert spaces;
- operator theory on analytic function spaces;
- spectral properties of individual operators;
- algebras generated by specific classes of operators.
- Operator Theoretical Methods in Mathematical Physics
*Luís Castro*, Universidade de Aveiro, Portugal*Frank-Olme Speck*, Instituto Superior Técnico, Portugal- Solution of boundary value and interface problems in mathematical physics. Typically elliptic problems in weak formulation are treated in Sobolev and related functional spaces by a systematic use of operator relations, local principles, normalization methods, and regularity theory.
- Operator factorization and operator relations in a theoretical sense. For instance, the equivalence or implication of different operator relations and their mutual construction is considered as an independent new subarea of operator theory as well as their usefulness in various applications.
- Special classes of pseudo-differential and Fourier integral operators. New prototypes of operators originated, e.g., from wedge diffraction problems, such as special Fourier integral operators or pure Hankel operators in the context of boundary value problems in conical Riemann surfaces. This involves also Banach algebra methods and the consideration of new functional spaces.

- Representation Theory of Algebras and Groups
*Carlos André*, Universidade de Lisboa, Portugal*Samuel Lopes*, Universidade do Porto, Portugal*Ana Paula Santana*, Universidade de Coimbra, Portugal- Spectral Theory and Differential Operators
*Andrii Khrabustovskyi*, Graz University of Technology, Austria*Olaf Post*, Universität Trier, Germany*Carsten Trunk*, Technische Universität Ilmenau, Germany- Toeplitz Operators, Convolution type Operators and Operator Algebras

Dedicated to Yuri Karlovich on his 70th Birthday *M. Amélia Bastos*, Instituto Superior Técnico, Portugal*Alexei Karlovich*, Universidade Nova de Lisboa, Portugal

Free noncommutative algebraic geometry, namely algebraic geometry where the variables range over square matrices of all sizes rather than over scalars as in the classical commutative case, goes back to the work of Amitsur in the 1960s, and is closely related to the general theory of free rings and their skew fields of fractions that has been developed by P.M. Cohn starting in the 1970s. During the last decade, it came into a fruitful contact with the topics of matrix and operator convexity and of free noncommutative function theory, with important motivations coming from the study of linear matrix inequalities (LMIs) in dimension independent optimization problems in systems and control. The goal of this special session is to bring together leading specialists, both senior and junior, in this exciting area at the crossroads of operator theory and noncommutative algebra.

In the first half of the 20th century, crucial scientific developments led by a host of re-searchers, such as Werner Heisenberg, John von Neumann, Wolfgang Pauli, David Hilbert and others made it clear that many important aspects of physics are accurately modeled by algebras of operators on Hilbert's space.

In many cases, observable quantities in the physical world are represented by linear operators satisfying certain algebraic relations which express fundamental physical principles. Properties of such objects and their relations are encoded in operator algebras. For example, Heisenberg's uncertainty principle states that the position and momentum of a particle cannot be accurately and simultaneously observed; this principle is famously expressed by the canonical commutation relation $PQ - QP = hI$; where $h$ is Planck's constant, and $P$ and $Q$ represent momentum and position, respectively.

The realization that algebras of operators should be brought into the study of these systems arises from the somewhat unexpected fact that it is often easier to understand the whole algebra generated by the operators of interest, rather than each individual operator by itself. Despite the fact that an algebra of operators is a much more formidable object, the tools of algebra and analysis have been shown to be extremely efficient in exploring the structure of these algebras, in turn shedding light on the operators from which they arose.

While the primary goal is understanding the generating operators arising from a given physical system, an excessive emphasis on the generators is rarely the best way to study the generated algebra. Instead, the most efficient method has been to apply ideas inspired by harmonic analysis, much like the study of locally compact abelian groups turns out to rely so fruitfully on its Pontryagin dual via the Fourier transform.

Among the most useful methods inspired by harmonic - analysis methods is based on certain dynamical systems arising from the algebraic structure. These dynamical systems may appear as the action of a group on a topological space, but they also arise as a semigroup, perhaps acting by means of partially defined maps, or even a pseudogroup. Regardless of its appearance, these dynamical systems often are codified by a groupoid.

The “Fourier analysis” part of the program consists of mapping the elements of the relevant algebra, or perhaps only a dense subalgebra, to functions on the appropriate groupoid. This is roughly analogous to the role played by the Pontryagin dual in Classical Harmonic Analysis. In turn the “spectral synthesis” part of the program consists in analyzing the algebraic structure from the point of view of the associated groupoid with the hope of reconstructing the original operator algebra, or at least constructing an operator algebra whose properties are closely reflected by the original.

The two main goals of the proposed special session are: a) to bring together a number of researchers whose work has contributed to the development of these exciting ideas and brought them to the forefront of operator algebras; and b) to allow young researchers the opportunity to interact with these people and to explore the promising new results in the area.

In this session new developments in the areas of free analysis and free probability theory, and in particular their interaction and their various applications, will be highlighted. Recent progress in this area has led to important new results in fields of mathematics as diverse as random matrix theory, quantum information theory, von Neumann algebras, or mathematical physics. Relevant topics are: free transport; the analysis of free SDEs; applications of free analysis to random matrix theory; or, conversely, applications of random matrix theory to operator theory and to quantum information theory.

During the last decade there has been an increasing interest on the application of methods of hypercomplex analysis in Operator Theory. Notable events were the development of a spectral theory for quaternionic operators based on the so called S-spectrum, the use reproducing kernel methods and Schur analysis in the hyperholomorphic setting.

This section has the aim to invite leading experts to present and disseminate the latest advances in this field.

There is a close and natural link between operator theory and the study of function spaces, and progress in one topic goes hand in hand with progress in the other. This is reflected in new developments highlighting this connection, as illustrated by recent works on the relations between Toeplitz or Wiener-Hopf type operators and various spaces of analytic functions such as model spaces, multipliers between Toeplitz kernels, Banach algebras, reflexivity and hyper-reflexivity, spectral theory of non self-adjoint operators, Riemann-Hilbert problems, and their applications in Mathematical Physics and Engineering.

Our aim is to bring together experts and researchers working on these and related topics to present the state of the art in their research, point out new research directions, and contribute to new advances and applications.

Early papers on Operator Theory focused on the spectral theory of a single self-adjoint operator on a Hilbert space with motivation and interpretation from the emerging theory of Quantum Mechanics from which emerged a complete spectral theorem and functional calculus (for bounded measurable functions on the spectrum of the operator) for self-adjoint operators as well as unitary operators and more generally normal operators. The next quest (well underway by the 1960s) was to understand classes of nonselfadjoint/nonunitary operators; early successes were the model theory of Livsic and de Branges-Rovnyak as well as that of Sz.-Nagy-Foias based on the notion of a unitary dilation for a contraction operator. The associated functon theory was for operator-valued holomorphic functions mapping the unit disk to contraction operators — as in the case of the Sz. Nagy-Foias characteristic function, or alternatively holomophic functions mapping the upper half plane into operators with positive imaginary part. Later work starting in the late 1970s with revived activity in the 1990s pushed the ideas to the setting of more restrictive classes of contraction operators ($n$-hypercontractions) with corresponding classes of holomorphic functions on the unit disk more restrictive than the Hardy class (specifically, weighted Bergman spaces). All these theories concern one operator at a time. The field of Multivariable Operator Theory seeks to understand parallel ideas for tuples of bounded linear operators rather than a single linear operator; the operator tuples may be commutative, or, at the opposite extreme, freely noncommutative. There are now emerging model theories and characteristic functions for $n$-hypercontractive commutative as well as freely noncommutative operator tuples, with associated function theory involving serveral-variable holomorphic functions on various types of domains in $\mathbb{C}^d$ for the commutative case, or free noncommutative functions on noncommutative-domain analogues of classical domains (e.g., bounded symmetric domains) in $\mathbb{C}^d$. The purpose of the session will be to bring together currently active researchers in this lively area of Multivariable Operator Theory for mutual communication on the current state of knowledge in the field.

The main goal of this Session is to present recent advances in the studies of main operators of harmonic analysis, which nowadays is one of the most active areas of Mathematical Analysis important for various kinds of applications.

Classically, operators of harmonic analysis are singular operators, potential-type operators and the maximal operators. More generally, they include also such operators as Hardy operators, hyper-singular operators, operators’ invariant with respect to dilation and rotation and others.

Last decades there was an increasing interest to the study of operators of harmonic analysis in the so called non-standard function spaces, such as variable exponent Lebesgue, Orlicz, Morrey, Hölder, Bergman spaces, their grand versions and others. This interest was caused both by applications in partial differential equations and challenges of mathematical difficulties in working in such spaces.

Topics of the Session may include studies not only in such non-standard function spaces but also in classical function spaces of measurable and smooth functions, both in real and complex variable settings. Various applications are welcome.

The aim of the session is to bring together researchers in this rapidly developing area, for interchange of knowledge and discussion of the current state of studies.

The session will be devoted, but not limited to the following topics:

The area of the session includes the following topics and related features.

Representation theory appears at the interface of many areas of Mathematics and Mathematical Physics, including (Lie) Group and Ring Theory, Combinatorics, Non-commutative (Algebraic) Geometry, Operator Algebras, Statistical Mechanics and many others. The aim of this special session is to bring together experts, students and researchers from related fields interested in the area of Representation Theory, reporting on the latest developments in this lively area and promoting discussions and further interactions.

Every operator has a secret treasure: its spectrum. You need a smart mathematician to hunt it. And treasure hunts are successful if one masters the tools from modern days operator theory. Any smart hunter knows that spectra may hide everywhere: in the rainbow colors and in the energies of mysterious quantum particles, in magnificent sounds of violin strings and beats of loud drums. Thus, the understanding of spectra theory leads us to understand nature.

It was the late Israel Gohberg — the first president of the IWOTA conference series — who said: "...beyond every good operator theory there is a concise equation...”. In many applications, the equation comes as ordinary or partial differential equation.

This point of view is the guiding motif of the special session Spectral Theory and Differential Equation within the IWOTA 2019 meeting in beautiful Lisboa, Portugal.

Every treasure hunter is more than welcome!

Toeplitz operators constitute one of the most important classes of non selfadjoint operators and their theory is interesting in itself but also has applications and connections with other areas like function theory, $C^\ast$ algebras, probability and physics. This session is mainly devoted to Toeplitz operators, convolution type operators and operator algebras generated by them. Many problems of the engineering sciences, physics, and mathematics lead to convolution operators and their various modifications.

The Fredholm theory of Toeplitz operators and Wiener-Hopf operators with continuous or piecewise continuous symbols is well understood. Recent results are focused on the Fredholm theory for algebras of convolution type operators with matrix-valued symbols admitting slow oscillations or oscillations of almost-periodic type; on less developed invertibility theory; and on closely related theory of factorization of matrix-valued symbols of convolution type operators. N. Krupnik demonstrated that does not exit a matrix symbol for the algebras containing singular integral operators whose coefficients have discontinuities of almost periodic type or non Carleman shifts operators and in order to solve this limitation the concept of operational symbol arises. In this direction recently faithful representations were constructed for $C^\ast$ algebras associated to $C^\ast$ dynamical systems with different groups of unitary operators indexed on amenable discrete groups of shifts. Finite sections and other approximation methods for algebras of convolution operators with piecewise continuous and oscillating symbols have been intensively studied as well. Applications to explicit solutions for infinite dimensional integrable systems such as the $KdV$ equation can be tackled by the factorization of the symbol of Toeplitz operators and recently some interesting results were achieved.

We plan to gather leading experts in the field with a goal to exchange the information on the latest achievements in the area, of which the referred are examples, and to discuss directions of further research on the subject.