Properties of a polyanalytic functional calculus on the S-spectrum

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy-Riemann operator in R-4, denoted by D. This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S-functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Delta, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of Delta=DD<overline>. Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator D<overline> and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of D-2. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.