On the Fundamental Principles of Unbounded Functional Calculi

Abstract: In this paper, a new axiomatization for unbounded functional calculi is proposed and the associated theory is elaborated comprising, among others, uniqueness and compatibility results and extension theorems of algebraic and topological nature. In contrast to earlier approaches, no commutativity assumptions need to be made about the underlying algebras. In a second part, the abstract theory is illustrated in familiar situations (sectorial operators, semigroup generators, normal operators). New topological extension theorems are proved for the sectorial calculus and the Hille--Phillips calculus. Moreover, it is shown that the Stieltjes and the Hirsch calculus for sectorial operators are subcalculi of a (small) topological extension of the sectorial calculus.