IWOTA 2019

International Workshop on
Operator Theory and its Applications

July 22-26, 2019
Instituto Superior Técnico
Lisbon, Portugal

# Abstracts

Ghent University

## Integration over manifolds of higher co-dimension embedded in $\mathbb{R}^m$ using distribution theory

There are two basic ways of describing smooth ($m-k$)-surfaces in $\mathbb{R}^m$. We either use a parametric definition as the image of a map from a subset of $\mathbb{R}^{m-k}$ to $\mathbb{R}^m$, or we use an implicit definition by means of equations $P_1(x)=\ldots=P_k(x)=0$. Most integration methods have traditionally been developed for a parametric definition. However, some methods of calculation are also necessary in the case where a ($m-k$)-surface is implicitly defined.

In the present talk, we illustrate a distributional approach to integration over embedded surfaces of any co-dimension $k < m$. This method follows from the link between differential forms and the Dirac distribution concentrated on manifolds. This approach enables an easy proof of a Cauchy formula for the tangential Dirac operator on a ($m-k$)-surface. In addition, we will apply this method to compute integrals over real Stiefel manifolds.

Daniel Alpay
Chapman University, Orange, CA, USA

## Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra

We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives.

This is joint work with Ismael Paiva and Daniele Struppa.

Maria Elena Luna Elizarraras
Holon Institute of Technology

## On some properties of the set of bicomplex numbers seen as a module over hyperbolic numbers

The versatile nature of the set $\operatorname{BC}$ of bicomplex numbers allows to see it as a real or a complex linear  space but also as a hyperbolic and a bicomplex module. It turned out that considering it as a module over hyperbolic numbers one deals with a rich and somewhat paradoxical object where the metric and the angles should be measured  with positive hyperbolic numbers thus arriving at the structure which is deeply similar to the usual complex plane with the usual Euclidean structure. All this and many nice consequences will be presented in the talk.

The talk is based on a joint work with A. Balankin and M. Shapiro.

Nelson José Rodrigues Faustino

## Bayesian Probability Formulation of Hypercomplex Fock States

This talk corresponds to an abridged version of the paper https://doi.org/10.1016/j.amc.2017.07.080. It addresses the construction of bound states to the discrete electromagnetic Schroedinger operator. Several interesting properties about the construction of the bound states are also highlighted in interplay with Bayesian probabilities, on which the Fock states arise as discrete quasi-probability distributions carrying a set of independent and identically distributed (i.i.d) random variables.

Several non-trivial examples of quasi-probability distributions will be presented with the aid of Mellin-Barnes integral representations for generalized Mittag-Leffler, Wright functions and alike, that in turn yield fractional counterparts for the Poisson and hypergeometric distributions as well.

One will also present some particular cases on which the underlying discrete Hamiltonian is isospectrally equivalent to a family of Appell sets of polynomials encoded by the orthosymplectic Lie algebra $\operatorname{osp}(1|2)$. In parallel, one will employ Dirac’s insight on negative probabilities to establish an intriguing fact on discrete quantum theory towards Macdonald–Ruijnaars approach: The ground state of the discrete Hamiltonian may be not to be a real-valued function, in general.

Milton Ferreira
Instituto Politécnico de Leiria & CIDMA, Universidade de Aveiro

## Fractional Borel-Pompeiu formula and a related hypercomplex fractional operator calculus

In this talk, we present a theory for operator calculus in Clifford analysis using fractional operators. We introduce Dirac operators of fractional order by using Riemann-Liouville and Caputo fractional derivatives and prove fractional Stokes' formula and Borel Bompeiu’s formula. These tools in hand together with a representation formula for the fundamental solution in terms of Mittag-Leffler operator series allow us to introduce generalizations of the Teodorescu and the Cauchy integral operator. Finally, we present a Hodge decomposition together with some applications to boundary value problems related to the fractional Laplacian.

Based on joint work with R. S. Kraußhar, M. M. Rodrigues, and N. Vieira.

Ren Hu
Ghent University

## Bargmann-Radon transform for axially monogenic functions

In our work we study a class of monogenic functions called axially monogenic functions. First we present the explicit form for the general Cauchy-Kowalewski extension for axially monogenic functions. Then we determine the Bargmann-Radon transform for these functions, relying on Funk-Hecke theorem in the process.

Dmitrii Legatiuk
Bauhaus University Weimar

## Interpolation of monogenic functions by using reproducing kernel Hilbert spaces

The main goal of this talk is to present results on the interpolation problem for monogenic functions in the unit ball in $\mathbb{R}^{d+1}$ by working with the reproducing kernel Hilbert space generated by the Bergman kernel. In this setting, we propose an algorithm which allows us to interpolate a monogenic function from a given countable number of observations (possible infinite) in the unit ball. We allow for the points to be randomly chosen in the ball but uniformly distributed and discuss the question of sparse reconstruction from only a few interpolation data. Theoretical results and illustrative examples will be presented.

Astrid Massé
Ghent University

## Discrete Heat Equation

Many research has been devoted to the study of the heat equation, as this partial differential equation finds applications in, among others, physics, probability theory and financial mathematics. Our aim is to discretize this equation in the one dimensional discrete Clifford framework. We will consider both discrete space and discrete time. We determine a fundamental solution and investigate its properties. Discrete heat polynomials are constructed: discrete polynomial solutions to the heat equation of degree n with the n-th basic discrete homogeneous polynomial as initial condition. The analogue problem in dual space, i.e. concerning distributions is discussed. We investigate the link with the discrete Weierstrass transform.

Teppo Mertens
UGent

## Radon-Type transforms for holomorphic functions in the Lie ball

In this talk we will discuss certain Radon-type transforms and their reproducing kernels which are studied in the paper by Sabadini and Sommen. Extending their work we will see some new results I found in joint work with F. Sommen.

Marco Peloso
Università degli Studi di Milano

## Fractional Bernstein spaces in several complex variables

In this talk I will present some recent results in the theory of function spaces of entire functions in several complex variables whose restriction to the boundary of the Siegel upper half space enjoy some integrability condition. The main tool is harmonic analysis on the Heisenberg group. This talk is based on joint work with A. Monguzzi and M. Salvatori.

Guangbin Ren
University of Science and Technology of China

## Function theory in slice analysis

We shall talk about some of our recent results on the function theory in slice analysis of quaternions. We find  that the canonical measure in the slice theory is the slice Lebesgue measure. This measure can be employed to define the Bergman spaces of slice regular functions. To study its dual and predual spaces we need to establish the Forelli-Rudin type estimates for some reproducing kernels.

We consider the boundary behavior of slice regular functions in the unit ball of quaternion and show that in its boundary Schwarz lemma the derivative of regular functions at the boundary fixed point is no longer real in general  and it keeps real only after  modification with a Lie bracket term reflecting the non-commutative feature of quaternions. We introduce a type of slice regular composition and initiate the study of the theory of composition operators. The point is that the resulting function after composition will not shrink its definition domain while it will do in the classical definition of composition due to the Bohr phenomena. We also study the geometric function theory and establish the sharp growth and distortion theorem as well as  the Bieberbach conjecture in the setting of slice functions.

Politecnico di Milano

## Perturbation of normal quaternionic operators

The theory of quaternionic operators has applications in several  different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis. In this talk, we discuss the problem of perturbation of a quaternionic normal operator in a Hilbert space, using the concepts of $S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators. We show results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and discuss conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

Joint work with P. Cerejeiras, F. Colombo, U. Kaehler.

Matteo Santacesaria
University of Genoa

## Calderon’s inverse problem and quaternionic analysis

Calderon’s problem consists in the determination of an electrical conductivity distribution inside a domain from boundary voltage and currents measurements, modeled by the Dirichlet-to-Neumann map. Complex analytic methods, including quasiconformal mappings and d-bar equations, have been extensively used to study the problem in two dimensions. Only few results have been obtained in higher dimensions with similar methods. One of the main questions is global uniqueness, that is the injectivity of the map that associates the boundary data to given conductivity. In two dimensions, this have been obtained by Astala-Paivarinta in 2006 for measurable conductivities, while in higher dimensions this is still an open conjecture.

In this talk I will give a review of the problem and present an approach based on quaternionic analysis to study Calderon’s problem in three dimensions for measurable conductivities.

Michael Shapiro
Holon Institute of Technology

## On the interplay between real and quaternionic structures on linear spaces

Given a real linear space, we analyze when and how it can be endowed with the structure of a quaternionic linear space. If the initial real space possesses some additional structures (topology, norm, inner product) the we extend the analysis onto this situation. As then next step we deal with a linear operator acting between two real linear spaces and we are interested in finding out when the operator can be made quaternionic linear if its domain and range were quaternionized. Some applications to the theory of Bochner and Pettis integrals will be given.

The talk is based on a joint work with J. O. González-Cervantes, M. E. Luna-Elizarrarás and F. Ramírez-Reyes.