@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-ZW34QRFX-C
  skos:broader psr:-KL7BX9Z3-T ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Resolvent_formalism>, <https://fr.wikipedia.org/wiki/R%C3%A9solvante> ;
  dc:modified "2023-08-04"^^xsd:date ;
  skos:prefLabel "resolvent formalism"@en, "résolvante"@fr ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:inScheme psr: ;
  skos:definition """In mathematics, the <b>resolvent formalism</b> is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus.
<br/>The <b>resolvent</b> captures the spectral properties of an operator in the analytic structure of the functional. Given an operator <span class="texhtml mvar" style="font-style:italic;">A</span>, the resolvent may be defined as
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle R(z;A)=(A-zI)^{-1}~.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>R</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo>;</mo>
<br/>        <mi>A</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>A</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>z</mi>
<br/>        <mi>I</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R(z;A)=(A-zI)^{-1}~.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71cf60c18da4995914e32ca463367faf76cfb54c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.75ex; height:3.176ex;" alt="R(z;A)=(A-zI)^{{-1}}~."> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Resolvent_formalism">https://en.wikipedia.org/wiki/Resolvent_formalism</a>)"""@en, """Soit <span class="texhtml"><i>A</i></span> un opérateur linéaire (non nécessairement continu) défini sur un espace de Banach. Pour tout nombre complexe <span class="texhtml">λ</span> tel que <span class="texhtml">(λ I – <i>A</i>)<sup>–1</sup></span> existe et est continu, on définit la <b>résolvante</b> de <span class="texhtml"><i>A</i></span> par&nbsp;:
<br/>
<br/><div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle R_{\\\\lambda }=(\\\\lambda \\\\mathrm {I} -A)^{-1}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>R</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>λ<!-- λ --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>λ<!-- λ --></mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">I</mi>
<br/>        </mrow>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>A</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R_{\\\\lambda }=(\\\\lambda \\\\mathrm {I} -A)^{-1}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7333ca67775718bd339f1643cb4e15ae16ef69" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.62ex; height:3.176ex;" alt="R_{\\\\lambda} = (\\\\lambda \\\\mathrm I- A)^{-1}."></span></div>
<br/>L'ensemble des valeurs de <span class="texhtml">λ</span> pour lesquelles la résolvante existe est appelé l'<b>ensemble résolvant</b>, noté <span class="texhtml">ρ(<i>A</i>)</span>. Le <i>spectre</i> <span class="texhtml">σ(<i>A</i>)</span> est le complémentaire de l'ensemble résolvant&nbsp;: <span class="texhtml">σ(<i>A</i>)</span> = ℂ \\\\ <span class="texhtml">ρ(<i>A</i>)</span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/R%C3%A9solvante">https://fr.wikipedia.org/wiki/R%C3%A9solvante</a>)"""@fr ;
  skos:related psr:-V80329XF-X ;
  a skos:Concept .

psr:-V80329XF-X
  skos:prefLabel "Fredholm integral equation"@en, "équation intégrale de Fredholm"@fr ;
  a skos:Concept ;
  skos:related psr:-ZW34QRFX-C .

psr:-KL7BX9Z3-T
  skos:prefLabel "opérateur"@fr, "operator"@en ;
  a skos:Concept ;
  skos:narrower psr:-ZW34QRFX-C .

psr: a skos:ConceptScheme .