The harmonic $H^{\infty }$-functional calculus based on the $S$-spectrum

Abstract

The aim of this paper is to introduce the $H^{\infty }$-functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to $Df(T)$ for unbounded sectorial operators $T$ and polynomially growing functions of the form $Df$, where $f$ is a slice hyperholomorphic function and $D={\partial }_{q_0}+e_1{\partial }_{q_1}+e_2{\partial }_{q_2}+e_3{\partial }_{q_3}$ is the Cauchy–Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known $S$-functional calculus $f(T)$, with a different resolvent operator. The harmonic $H^{\infty }$-functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator $Df(T)$ directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators $Df(T)$. The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the $S$-spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter–Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.