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Generation of generators of holomorphic semigroups

Part of: Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Christian Berg ,
Khristo Boyadzhiev  and
Ralph Delaubenfels
Christian Berg
Affiliation:
Matematisk Institut Universitetsparken, 5, DK-2100, Kobenhavnø, Demark
Khristo Boyadzhiev
Affiliation:
Ohio Northern University, Ada, Ohio 45810, USA
Ralph Delaubenfels
Affiliation:
Ohio University, Athens, Ohio 45701, USA
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Abstract

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We 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 functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.

We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/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.

MSC classification

Secondary: 47A60: Functional calculus 47B44: Accretive operators, dissipative operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 246 - 269
Copyright
Copyright © Australian Mathematical Society 1993

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