The -spectrum and the -functional calculus

Abstract

In some recent papers (called -functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The -functional calculus is based on the notion of -spectrum, which naturally arises from the definition of the -resolvent operator for n-tuples of operators. The -resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the -spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the -spectrum. In the case of commuting operators, our functional calculus is based on the -spectrum and will be called -functional calculus. We point out that for a correct definition of the -resolvent operator and of the -resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the -spectrum is that it is related to the -functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.