# Functional calculi, regularized semigroups and integrated semigroups

Ralph deLaubenfels; Mustapha Jazar

Studia Mathematica (1999)

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

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topdeLaubenfels, 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 -

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