- 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 S-functional calculus for quaternionic operators using the notion of the S-spectrum. 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 well-known Fueter-Sce-Qian mapping theorem in the polyanalytic case. A main goal would be to introduce a functional calculus for poly-monogenic 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 Clifford-Appell system approach. To this end, I will need to use Dirac operators, monogenic and slice monogenic functions, the S-functional calculus and spectral analysis techniques. Finally, I will study two new type of Fock spaces and related integral transforms such as the Segal-Bargmann 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 Leffler-Fock kernel.