Generation of generators of holomorphic semigroups

Abstract

AbstractWe construct a functional calculus,g→g(A), for functions,g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators,A, whose resolvent set contains (−∞, 0), with {‖r(r+A)−1‖ ¦r> 0} bounded. For such functionsg, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.We show that, for any Bernstein functionf, −f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever −Adoes.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.