On the Taylor functional calculus (Q2773430)

Let \(A=(A_1, \dots, A_n)\) be an \(n\)-tuple of mutually commuting operators acting on a Banach space. It is pointed out that the known representation of \(f(A)\) for \(f\) analytic on a neighbourhood of the Taylor spectrum of \(A\) is rather complicated. In this connection the author gives a new Martinelly-Vasilescu type formula for \(f(A)\) and uses it to obtain simple proofs of basic properties of the Taylor functional calculus, such as the superposition property and the spectral mapping theorem.