# Journal of Operator Theory

Volume 88, Issue 2, Fall 2022  pp. 309-364.

Poly slice monogenic functions, Cauchy formulas and the $PS$-functional calculus

Authors:  Daniel Alpay (1), Fabrizio Colombo (2), Kamal Diki (3), Irene Sabadini (4)
Author institution:(1) Faculty of Mathematics Physics and Computation,\break Schmid College of Science and Technology, Chapman University, One University Drive Orange, California 92866, U.S.A.
(2) Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9 20133 Milano, Italy
(3) Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9 20133 Milano, Italy
(3) Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9 20133 Milano, Italy

Summary: Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the $S$-spectrum have had a very fast development. This new spectral theory based on the $S$-spectrum has applications, for example, in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. In this paper we introduce and study the theory of poly slice monogenic functions, also proving some Cauchy-type integral formulas. Then we introduce the associated functional calculus, called $PS$-functional calculus, which is the polyanalytic version of the $S$-functional calculus and which is based on the notion of $S$-spectrum. We study some different formulations of the calculus and we prove some of its properties, among which the product rules. % An abstract of at most 100 words and of about 8 rows should be included. % The abstract should summerise the results of the paper and nothing more. % The main purpose of the abstract is to enable the readers to take in the % nature and results of the papers quickly and to decide whether they are % willing to read the entire paper or not. Citation of bibliography within the % abstract should be avoided. Do not use custom macros inside the abstract.

DOI: http://dx.doi.org/10.7900/jot.2021feb20.2347
Keywords: poly slice monogenic function, Cauchy formulas, $PS$-resolvent operators, modified $S$-resolvent operators, spectral theory on the $S$-spectrum