Functional Calculus for the Series of Semigroup Generators Via Transference

Abstract

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved thatif - be the sequence generates 0 -semigroups on a Hilbert space, then for each > -1 the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half-plane. The bounded of this calculus grows at most logarithmically as(1 + ) ↘ 0. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a socalled (strong) m-bounded calculus for all m ∈ ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called - semigroups, the Hilbert space results actually hold in general Banach spaces.