Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if −A   generates a C0C0-semigroup on a Hilbert space, then for each τ>0τ>0 the operator A   has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy f(z)=O(e−τRe(z))f(z)=O(e−τRe(z)) as |z|→∞|z|→∞. The bound of this calculus grows at most logarithmically as τ↘0τ↘0. As a consequence, f(A)f(A) is a bounded operator for each holomorphic function f (on a right half-plane) with polynomial decay at ∞. Then we show that each semigroup generator has a so-called (strong) m  -bounded calculus for all m∈Nm∈N, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called γ-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.