Domination for subordinated positive semigroups

We first give a necessary and sufficient condition for -f(B) to be dominated by -f(A), where f is a completely Bernstein function, B and A are C0-semigroup generators. Then we prove that there is no domination relationship between a semigroup and the subordinated semigroup if additional conditions are satisfied.

Functional Calculus for the Series of Semigroup Generators Via Transference

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 that if be the sequence generates 0- semigroups on a Hilbert space, then for each 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. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (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.