On some notions of convergence for n-tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm-resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S-resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S-functional calculus for n-tuples of noncommuting operators. The same notion can be extended to the case of the F-resolvent operator, which is the basis of the F-functional calculus, a monogenic functional calculus for n-tuples of commuting operators. We also prove some properties of the F-functional calculus, which are of independent interest.