We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them. A ∋ f → f (A) := 1 2πi Γ f (λ)(λ − A) −1 dλ,

We prove that for any Bernstein function ψ the operator −ψ(A) generates a bounded holomorphic C 0 -semigroup (e −tψ(A) ) t≥0 on a Banach space, whenever −A does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that (e −tψ(A) ) t≥0 is holomorphic in the holomorphy sector of (e −tA ) t≥0 , and if (e −tA ) t≥0 is sectorially bounded in this sector then (e −tψ(A) ) t≥0 has the same property. We also obtain new sufficient conditions on ψ in order that, for every Banach space X, the semigroup (e −tψ(A) ) t≥0 on X is holomorphic whenever (e −tA ) t≥0 is a bounded C 0 -semigroup on X. These conditions improve and generalize well-known results by Carasso-Kato and Fujita.

We consider the problem of estimates for the powers of the Cayley transform V = ( )( ) A I A I + − −1 of the generator of a uniformly bounded C 0 -semigroup of operators e tA , t ≥ 0, that acts in a Hilbert space H . In particular, we establish the estimate sup ln( ) / n n V n ∈ + < ∞ ( ) N 1 . We show that the estimate sup n n V ∈ < ∞ N is true in the following cases: (a) the semigroups e tA and e tA −1 are uniformly bounded; (b) the semigroup e tA uniformly bounded for t ≥ ∞ is analytic (in particular, if the generator of the semigroup is a bounded operator).

We create a new, functional calculus, approach to approximation of C0-semigroups on Banach spaces. As an application of this approach, we obtain optimal convergence rates in classical approximation formulas for C0-semigroups. In fact, our methods allow one to derive a number of similar formulas and equip them with sharp convergence rates. As a byproduct, we prove a new interpolation principle leading to efficient norm estimates in the Banach algebra of Laplace transforms of bounded measures on the semi-axis.