Towards a general [formula omitted]-resolvent equation and Riesz projectors

Abstract

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to write the Fueter-Sce-Qian extension theorem in integral form and to define the F-functional calculus for n-tuples of commuting operators. This functional calculus is defined on the S-spectrum and generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the F-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the F-resolvent equation which was previously known only in the quaternionic setting and also its existence was under question. In this paper we prove the F-resolvent equation in the Clifford algebra setting. It is much more complicated than the one in the quaternionic case since it contains various pieces, however it still allows to nicely define the Riesz projectors.