Integrated semigroups and \(C\)-semigroups and their applications
type: Item
instance of: scholarly article
title: Integrated semigroups and \(C\)-semigroups and their applications
zbMATH Open document ID: 1471.47030
author: N. Yu. Selivanova
author: V. V. Vasil'yev
author: S. I. Piskarev
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-06-21T00:00:00Z
review text: The survey is devoted to recent advances in integrated semigroups and $C$-semigroups of operators in Banach space and their applications to the regularization of ill-posed problems. Typically all theorems, propositions, etc., are given with relevant references and without proofs. \par Chapter 1 concerns $n$-times integrated semigroups on Banach spaces. General properties of such type of operator semigroups and the relationship to abstract Cauchy problems are discussed. The generator of an integrated semigroup is defined and its (spectral) properties are described. It is emphasized that in the general case the generator may be not densely defined. \par Special attention is paid to the case of exponentially bounded integrated semigroups. The authors review results on relations between the semigroup, the resolvent of the generator, and the spectral properties of the generator. Interpolation results are given as well. It is considered the particular case of local integrated semigroups and their properties. \par Chapter 2 concerns $C$-semigroups. The first part of Chapter~2 is devoted to exponentially bounded semigroups. The authors give the definitions of the generator of a semigroup and the generators in the sense of Miyadera and in the sense of Da Prato. They discuss the relations between different notions of the semigroup generators and their spectral properties. The relations between semigroups and resolvents are discussed as well. Some interpolation results for the semigroup generator are given. The case of analytic semigroups is considered. The relations with abstract Cauchy problems and basic perturbation results are considered as well. \par The second part of Chapter~2 concerns local $C$-semigroups and $C$-cosine operator functions. Basic notions and results on local $C$-semigroups (i.e., generator, asymptotic $C$-resolvent, etc.)\ are discussed. General results on local $C$-cosine families and the relations with abstract second-order Cauchy problems are given. …
reviewed by: Nikita V. Artamonov
Mathematics Subject Classification ID: 47D60
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 65F22
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 6892358
zbMATH Keywords: semigroup
zbMATH Keywords: integrated semigroup
zbMATH Keywords: C-semigroup
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
full work available at URL: s10958-018-3760-x
OpenAlex ID: W2790904492
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Integrated semigroups and \(C\)-semigroups and their applications (cites work)
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Integrated semigroups and \(C\)-semigroups and their applications (Wikidata QID)
Wikidata QID: Q130067277

Integrated semigroups and \(C\)-semigroups and their applications (DOI)
DOI: 10.1007/S10958-018-3760-X