
Natural sciences
 Functional analysis ^{ }
 Functions of a complex variable ^{ }
 Integral transforms, operational calculus ^{ }
 Operator theory ^{ }
 Several complex variables and analytic spaces ^{ }
In this project, I will investigate research problems of current interest in (hyper)complex analysis and operator theory. I will study the polyanalytic counterpart of the Sfunctional calculus for quaternionic operators using the notion of the Sspectrum. One of the most important results that I want to prove in this framework is the polyanalytic counterpart of the spectral mapping theorem which will lead to interesting applications in operator theory. I will investigate also extension, inverse problem and applications of the wellknown FueterSceQian mapping theorem in the polyanalytic case. A main goal would be to introduce a functional calculus for polymonogenic functions using an integral representation of this result. Then, I will study also complex and quaternionic reproducing kernel Hilbert spaces and related operators with respect to various function theories. In particular, I will be interested by the complex polyanalytic functions of infinite order, the slice polyanalytic functions and the CliffordAppell system approach. To this end, I will need to use Dirac operators, monogenic and slice monogenic functions, the Sfunctional calculus and spectral analysis techniques. Finally, I will study two new type of Fock spaces and related integral transforms such as the SegalBargmann and Berezin transforms. The first space is inspired by the Gaussian RBF(radial basis function) kernel, while the second one is built upon a Mittag LefflerFock kernel.