From the bounded \(S\)-functional calculus to the \(H^\infty\)-functional calculus

The quaternionic (or Clifford algebra) version of the Cauchy integral formula motivates the so called bounded \(S\)-functional calculus, i.e., we formally replace the variable which is not integrated in the Cauchy integral formula by a bounded operator \(T\).
For bounded operators, this integral makes sense, since the spectrum of \(T\) is bounded and the integration path is compact.

For unbounded, closed operators \(T\), there are different generalizations of the functional calculus. In particular, for the important class of sectorial operators, this talk will introduce the \(H^\infty\)-functional calculus in a 2-step procedure. First, functions \(f\) with a certain decay at infinity are considered, enough decay such that the integral still makes sense. In a second step, using a regularization procedure, \(f(T)\) is defined also for polynomially growing functions.