On the formulations of the quaternionic functional calculus

In the paper [F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009) 601–627] the authors treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute. This functional calculus is the quaternionic version of the classical Riesz–Dunford functional calculus. When considering quaternionic operators it is natural to also consider the case of left linear operators. Furthermore, one can use left or right slice regular functions to construct a functional calculus for right (or left) linear operators. In this paper we discuss these possibilities, showing that, in all the cases, we can associate to an operator two so-called SS-resolvent operators but their interpretation depends on whether we are considering a left or a right linear operator. Also the SS-resolvent equations for right or left closed operators do not have the same interpretation. Moreover, we study the bounded perturbations of both the SS-resolvent operators.