Functional calculus for generations of uniformly bounded holomorphic semigroups (Q1824841)

On a Banach space one considers a linear operator A that generates a uniformly bounded holomorphic semigroup. The author's method of constructing a functional calculus of A uses the Poisson integral representation. On this way, necessary and sufficient conditions for A to be ``well-bounded'' or ``C\({}^ 1\)-scalar'' are given in terms of the family of operators \[ G(s,A)=\int_{{\mathbb{R}}}[1-\cos (sr)]e^{ir A}(dr/\pi r^ 2),\quad s\geq 0. \] As an application one shows that d/d\(\theta\) on \(H^ 1\) of the unit circle is \(C^ 1\)-scalar.