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 R4$\mathbb {R}^4$, denoted by D$\mathcal {D}$. 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, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of Δ=DD¯$\Delta = \mathcal {D} {\overline{\mathcal D}}$. Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator D¯$ {\overline{\mathcal D}}$ and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of D2$ \mathcal {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.