The F-functional calculus for unbounded operators

In the recent years the theory of slice hyperholomorphic functions has become an important tool to study two functional calculi for n-tuples of operators and also for its applications to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphic functions, it is possible to give the Fueter-Sce mapping theorem an integral representation. With this integral representation it has been defined a monogenic functional calculus for n-tuples of bounded commuting operators, the so called F-functional calculus. In this paper we show that it is possible to define this calculus also for n-tuples containing unbounded operators and we obtain an integral representation formula analogous to the one of the Riesz-Dunford functional calculus for unbounded operators acting on a complex Banach space. As we will see, it is not an easy task to provide the correct definition of the F-functional calculus in the unbounded case. This paper is addressed to a double audience, precisely to people with interests in hypercomplex analysis and also to people working in operator theory. (C) 2014 Elsevier B.V. All rights reserved.