Functional calculi, regularized semigroups and integrated semigroups

Ralph deLaubenfels; Mustapha Jazar

Studia Mathematica (1999)

  • Volume: 132, Issue: 2, page 151-172
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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deLaubenfels, Ralph, and Jazar, Mustapha. "Functional calculi, regularized semigroups and integrated semigroups." Studia Mathematica 132.2 (1999): 151-172. <http://eudml.org/doc/216592>.

@article{deLaubenfels1999,
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.},
author = {deLaubenfels, Ralph, Jazar, Mustapha},
journal = {Studia Mathematica},
keywords = {closed linear operators; abstract Cauchy problem; polynomially bounded solution; functional calculi; regularized semigroups; Laplace transform},
language = {eng},
number = {2},
pages = {151-172},
title = {Functional calculi, regularized semigroups and integrated semigroups},
url = {http://eudml.org/doc/216592},
volume = {132},
year = {1999},
}

TY - JOUR
AU - deLaubenfels, Ralph
AU - Jazar, Mustapha
TI - Functional calculi, regularized semigroups and integrated semigroups
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 2
SP - 151
EP - 172
AB - 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.
LA - eng
KW - closed linear operators; abstract Cauchy problem; polynomially bounded solution; functional calculi; regularized semigroups; Laplace transform
UR - http://eudml.org/doc/216592
ER -

References

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