In this talk some basics of the theory of functional calculus will be explained and attention is paid to some recent results in this area.
Suppose the Abstract Cauchy Problem associated to an operator A on a Banach space X is well-posed. Functional calculus theory provides a method of defining, for f a holomorphic function on a right half-plane, an operator f(A) on X. It then aims to study the correspondence f->f(A) and to relate properties of f(A) to those of f. In many settings norm bounds for f(A) are important, for instance in the analysis of numerical methods of approximation. The definition of f(A) is first made for a class of elementary functions. A transference method is a manner of factorizing these elementary operators via convolution operators on suitable function spaces. The theory of Fourier multipliers is then used to derive norm bounds for the elementary operators. Finally, one approximates general functions by elementary ones. These transference methods link harmonic analysis to functional calculus theory and yield new results.
This is joint work with Markus Haase.