The theory of Besov functional calculus: developments and applications to semigroups

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra $\mathcal{B}$ of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of $\mathcal{B}$ and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory.