Fractional powers of generators of equicontinuous semigroups and fractional derivatives
Identifieur interne : 002569 ( Istex/Corpus ); précédent : 002568; suivant : 002570Fractional powers of generators of equicontinuous semigroups and fractional derivatives
Auteurs : Oscar E. Lanford ; Derek W. RobinsonSource :
- Journal of the Australian Mathematical Society [ 1446-7887 ] ; 1989-06.
Abstract
We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, Hα Hβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).
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DOI: 10.1017/S1446788700030950
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Lanford</name><affiliation><mods:affiliation>IHES 91440 Bures-sur-Yvette, France</mods:affiliation></affiliation></author><author><name sortKey="Robinson, Derek W" sort="Robinson, Derek W" uniqKey="Robinson D" first="Derek W." last="Robinson">Derek W. Robinson</name><affiliation><mods:affiliation>Mathematics Department, Institute of Advanced Studies, Australian National University, Canberra, Australia</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of the Australian Mathematical Society</title><title level="j" type="abbrev">J. Aust. Math. Soc.</title><idno type="ISSN">1446-7887</idno><idno type="eISSN">1446-8107</idno><imprint><publisher>Cambridge University Press</publisher><pubPlace>Cambridge, UK</pubPlace><date type="published" when="1989-06">1989-06</date><biblScope unit="volume">46</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="473">473</biblScope><biblScope unit="page" to="504">504</biblScope></imprint><idno type="ISSN">1446-7887</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1446-7887</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, Hα Hβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).</div></front></TEI><istex><corpusName>cambridge</corpusName><author><json:item><name>Oscar E. Lanford III</name><affiliations><json:string>IHES 91440 Bures-sur-Yvette, France</json:string></affiliations></json:item><json:item><name>Derek W. Robinson</name><affiliations><json:string>Mathematics Department, Institute of Advanced Studies, Australian National University, Canberra, Australia</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>47 A 99</value></json:item></subject><articleId><json:string>03095</json:string></articleId><arkIstex>ark:/67375/6GQ-9TN0F2WV-C</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>research-article</json:string></originalGenre><abstract>We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, Hα Hβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).</abstract><qualityIndicators><score>8.704</score><pdfWordCount>8948</pdfWordCount><pdfCharCount>45355</pdfCharCount><pdfVersion>1.4</pdfVersion><pdfPageCount>32</pdfPageCount><pdfPageSize>442.8 x 650.88 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractWordCount>142</abstractWordCount><abstractCharCount>733</abstractCharCount><keywordCount>1</keywordCount></qualityIndicators><title>Fractional powers of generators of equicontinuous semigroups and fractional derivatives</title><pii><json:string>S1446788700030950</json:string></pii><genre><json:string>research-article</json:string></genre><host><title>Journal of the Australian Mathematical Society</title><language><json:string>unknown</json:string></language><issn><json:string>1446-7887</json:string></issn><eissn><json:string>1446-8107</json:string></eissn><publisherId><json:string>JAZ</json:string></publisherId><volume>46</volume><issue>3</issue><pages><first>473</first><last>504</last><total>32</total></pages><genre><json:string>journal</json:string></genre></host><ark><json:string>ark:/67375/6GQ-9TN0F2WV-C</json:string></ark><publicationDate>1989</publicationDate><copyrightDate>1989</copyrightDate><doi><json:string>10.1017/S1446788700030950</json:string></doi><id>CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/fulltext/zip</uri></json:item><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/fulltext/txt</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Fractional powers of generators of equicontinuous semigroups and fractional derivatives</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher scheme="https://publisher-list.data.istex.fr">Cambridge University Press</publisher><pubPlace>Cambridge, UK</pubPlace><availability><licence><p>Copyright © Australian Mathematical Society 1989</p></licence><p scheme="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-G3RCRD03-V">cambridge</p></availability><date>1989</date></publicationStmt><notesStmt><note type="research-article" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="journal" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Fractional powers of generators of equicontinuous semigroups and fractional derivatives</title><author xml:id="author-0000"><persName><forename type="first">Oscar E.</forename><surname>Lanford</surname></persName><roleName type="degree">III</roleName><affiliation>IHES 91440 Bures-sur-Yvette, France</affiliation></author><author xml:id="author-0001"><persName><forename type="first">Derek W.</forename><surname>Robinson</surname></persName><affiliation>Mathematics Department, Institute of Advanced Studies, Australian National University, Canberra, Australia</affiliation></author><idno type="istex">CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6</idno><idno type="ark">ark:/67375/6GQ-9TN0F2WV-C</idno><idno type="DOI">10.1017/S1446788700030950</idno><idno type="PII">S1446788700030950</idno><idno type="article-id">03095</idno></analytic><monogr><title level="j">Journal of the Australian Mathematical Society</title><title level="j" type="abbrev">J. Aust. Math. Soc.</title><idno type="pISSN">1446-7887</idno><idno type="eISSN">1446-8107</idno><idno type="publisher-id">JAZ</idno><imprint><publisher>Cambridge University Press</publisher><pubPlace>Cambridge, UK</pubPlace><date type="published" when="1989-06"></date><biblScope unit="volume">46</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="473">473</biblScope><biblScope unit="page" to="504">504</biblScope></imprint></monogr></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1989</date></creation><langUsage><language ident="en">en</language></langUsage><abstract style="normal"><p>We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, Hα Hβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).</p></abstract><textClass><keywords scheme="keyword"><list><head>1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision)</head><item><term>47 A 99</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1989-06">Published</change></revisionDesc></teiHeader></istex:fulltextTEI></fulltext><metadata><istex:metadataXml wicri:clean="corpus cambridge not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="US-ASCII"</istex:xmlDeclaration><istex:docType PUBLIC="-//NLM//DTD Journal Publishing DTD v2.2 20060430//EN" URI="journalpublishing.dtd" name="istex:docType"></istex:docType><istex:document><article dtd-version="2.2" article-type="research-article"><front><journal-meta><journal-id journal-id-type="publisher-id">JAZ</journal-id><journal-title>Journal of the Australian Mathematical Society</journal-title><abbrev-journal-title>J. Aust. Math. Soc.</abbrev-journal-title><issn pub-type="ppub">1446-7887</issn><issn pub-type="epub">1446-8107</issn><publisher><publisher-name>Cambridge University Press</publisher-name><publisher-loc>Cambridge, UK</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.1017/S1446788700030950</article-id><article-id pub-id-type="pii">S1446788700030950</article-id><article-id pub-id-type="publisher-id">03095</article-id><title-group><article-title>Fractional powers of generators of equicontinuous semigroups and fractional derivatives</article-title><alt-title alt-title-type="left-running">Oscar E. Lanford III and Derek W. Robinson</alt-title><alt-title alt-title-type="right-running">Fractional powers of generators</alt-title></title-group><contrib-group><contrib><name><surname>Lanford</surname><given-names>Oscar E.</given-names><suffix>III</suffix></name><xref ref-type="aff" rid="aff1"></xref></contrib><contrib><name><surname>Robinson</surname><given-names>Derek W.</given-names></name><xref ref-type="aff" rid="aff2"></xref></contrib></contrib-group><aff id="aff1"><addr-line>IHES 91440 Bures-sur-Yvette</addr-line>, <country>France</country></aff><aff id="aff2"><addr-line>Mathematics Department</addr-line>, <institution>Institute of Advanced Studies</institution>, <institution>Australian National University</institution>, <addr-line>Canberra</addr-line>, <country>Australia</country></aff><pub-date pub-type="ppub"><month>06</month><year>1989</year></pub-date><volume>46</volume><issue>3</issue><fpage seq="15">473</fpage><lpage>504</lpage><history><date date-type="received"><day>21</day><month>09</month><year>1987</year></date></history><permissions><copyright-statement>Copyright © Australian Mathematical Society 1989</copyright-statement><copyright-year>1989</copyright-year><copyright-holder>Australian Mathematical Society</copyright-holder></permissions><abstract abstract-type="normal"><title>Abstract</title><p>We analyze fractional powers <italic>H</italic><sup>α</sup>, α > 0, of the generators <italic>H</italic> of uniformly bounded locally equicontinuous semigroups <italic>S</italic>. The <italic>H</italic><sup>α</sup> are defined as the αth derivative δ<sup>α</sup> of the Dirac measure δ evaluated on <italic>S</italic>. We demonstrate that the <italic>H</italic><sup>α</sup> are closed operators with the natural properties of fractional powers, for example, <italic>H</italic><sup>α</sup> <italic>H</italic><sup>β</sup> = <italic>H</italic><sup>α+β</sup> for α, β > 0, and (<italic>H</italic><sup>α</sup>)<sup>β</sup> = <italic>H</italic><sup>αβ</sup> for 1 > α > 0 and β > 0. We establish that <italic>H</italic>α can be evaluated by the Balakrishnan-Lions-Peetre algorithm <disp-formula><graphic xlink:href="S1446788700030950_eqn1" mime-subtype="gif"></graphic></disp-formula> where <italic>m</italic> is an integer larger than α, <italic>C<sub>α</sub>, m</italic> is a suitable constant, and the limit exists in the appropriate topology if, and only if, <italic>x ∈ D(H<sup>α</sup>)</italic>. Finally we prove that <italic>H</italic><sup>∈</sup> is the fractional derivation of <italic>S</italic> in the sense <disp-formula><graphic xlink:href="S1446788700030950_eqn2" mime-subtype="gif"></graphic></disp-formula> where the limit again exists if, and only if, <italic>x</italic> ∈ D(H<sup>α</sup>).</p></abstract><kwd-group kwd-group-type="mathsclassification"><title>1980 Mathematics subject classification (Amer. Math. Soc.) 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Robinson</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Fractional powers of generators of equicontinuous semigroups and fractional derivatives</title></titleInfo><name type="personal"><namePart type="given">Oscar E.</namePart><namePart type="family">Lanford</namePart><namePart type="termsOfAddress">III</namePart><affiliation>IHES 91440 Bures-sur-Yvette, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Derek W.</namePart><namePart type="family">Robinson</namePart><affiliation>Mathematics Department, Institute of Advanced Studies, Australian National University, Canberra, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="research-article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Cambridge University Press</publisher><place><placeTerm type="text">Cambridge, UK</placeTerm></place><dateIssued encoding="w3cdtf">1989-06</dateIssued><copyrightDate encoding="w3cdtf">1989</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><abstract type="normal">We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, Hα Hβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).</abstract><subject><genre>1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision)</genre><topic>47 A 99</topic></subject><relatedItem type="host"><titleInfo><title>Journal of the Australian Mathematical Society</title></titleInfo><titleInfo type="abbreviated"><title>J. Aust. Math. Soc.</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><identifier type="ISSN">1446-7887</identifier><identifier type="eISSN">1446-8107</identifier><identifier type="PublisherID">JAZ</identifier><part><date>1989</date><detail type="volume"><caption>vol.</caption><number>46</number></detail><detail type="issue"><caption>no.</caption><number>3</number></detail><extent unit="pages"><start>473</start><end>504</end><total>32</total></extent></part></relatedItem><identifier type="istex">CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6</identifier><identifier type="ark">ark:/67375/6GQ-9TN0F2WV-C</identifier><identifier type="DOI">10.1017/S1446788700030950</identifier><identifier type="PII">S1446788700030950</identifier><identifier type="ArticleID">03095</identifier><accessCondition type="use and reproduction" contentType="copyright">Copyright © Australian Mathematical Society 1989</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-G3RCRD03-V">cambridge</recordContentSource><recordOrigin>Copyright © Australian Mathematical Society 1989</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/metadata/json</uri></json:item></metadata><annexes><json:item><extension>gif</extension><original>true</original><mimetype>image/gif</mimetype><uri>https://api.istex.fr/document/CA721C6BBC48E8CA8EB82B990A11C89C201AB5A6/annexes/gif</uri></json:item></annexes><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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