Generation of generators of holomorphic semigroups

Abstract

We construct a functional calculus, g H> 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 (-oo, 0), with [\\r(r + A)' 1 1| | r > 0} bounded. For such functions g, we show that -g(A) generates a bounded holomorphic strongly continuous semigroup of angle 6, whenever -A does.We show that, for any Bernstein function f,-f(A) generates a bounded holomorphic strongly continuous semigroup of angle 7r/2, whenever -A does.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.1991 Mathematics subject classification (Amer. Math. Soc): 47 A 60, 47 B 44, 47 D 05.

Subordination in the sense of Bochner and a related functional calculus

“…[2] Subordination in the sense of Bochner and a functional calculus 369 (0.2) jx,(x) = I e~s x n , ( d s ) = etfM , t , x > 0 , J[0,oc) and is given by the Levy-Khinchine-type formula …”

“…It is well-known for contractive semigroups {7",},> 0 and one-sided a-stable subordinators f(x) = x a , a e (0, 1), that A f is indeed the fractional power -(-A)" (in the sense of Balakrishnan), see [23,13]. In [ 11,2,19] the complete Bernstein functions, a sufficiently rich subclass of the Bernstein functions (containing, for example, the above fractional powers), was considered and the relation A s = -/(-A) established whenever / is defined on the spectrum of -A, see Proposition 1.3. Here, -/(-A) is characterized via its resolvent in terms of the Dunford-Taylor integral, see Section 4 and [2,19].…”

Generation of Subordinated Holomorphic Semigroups via Yosida’s Theorem

“…Incidentally, it also partially answers the question from [3] and shows that Bernstein functions map the class of generators of sectorially bounded holomorphic C 0 -semigroups into itself. The statement was proved in [8] by a different technique.…”

“…Finally, we note that it is possible to develop an approach to the permanence problems from [12] and [3] different from the ones in [8] and in the present note. This approach based on direct resolvent estimates for Bernstein functions of semigroup generators is worked out in [2].…”

“…Moreover, it was proved in [3, Theorem 6.1 and Remark 6.2] that the above claim is true with no restrictions on θ ∈ (0, π/2] if a Bernstein function ψ is, in addition, complete. It was asked in [3] whether this additional assumption can, in fact, be removed.…”