A natural L∞ functional calculus for an absolutely continuous contraction is investigated. It is harmonic in the sense that for such a contraction and any bounded measurable function φ on the circle, the image can rightly be considered as φ̂(T ), where φ̂ is the solution of the Dirichlet problem for the disk with boundary values φ. The main result shows that if the functional calculus is isometri...

This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are essentials in Quantum Mechanics: unbounded operators, multiplication operators, self-adjointness, spectrum, functional calculus, spectral measures and von Neumann’s Spectral Theorem. 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . ...

The physicist R. Feynman advanced an idea of forming an abstract operator calculus for a finite number of noncommuting operators by attaching time indices to the operators. In joint work with G.W. Johnson, time-ordering measures are introduced to serve as weights for various functional calculi, such as the Weyl functional calculus (equal weights), the Kohn-Nirenberg or ordered functional calcul...

We introduce a notion of analytic generator for groups of unbounded operators, on Banach modules, arising from Esterle’s quasimultiplier theory. Characterizations of analytic generators are given in terms of the existence of certain functional calculi. This extends recent results about C0 groups of bounded operators. The theory is applicable to sectorial operators, representations of H∞, and in...