Functional calculi, regularized semigroups and integrated semigroups

Volume 132 / 1999

Ralph de Laubenfels, Mustapha Jazar Studia Mathematica 132 (1999), 151-172 DOI: 10.4064/sm-132-2-151-172

Abstract

We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of $A^n$, for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the right half-plane. We construct a semigroup analogue of a spectral distribution for such operators, and an extended functional calculus: When the abstract Cauchy problem has a unique $O(1 + t^k)$ solution for all initial data in the domain of $A^n$, for some nonnegative integer n, then a closed operator f(A) is defined whenever f is the Laplace transform of a derivative of any order, in the sense of distributions, of a function F such that $t → (1 + t^k)F(t)$ is in $L^{1}([0,∞))$. This includes fractional powers. In general, A is neither bounded nor densely defined.

Authors

  • Ralph de Laubenfels
  • Mustapha Jazar

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