Resolvent formalism

<h2 id="history">History</h2>
The first major use of the resolvent operator as a series in A (cf. <a href="/facts/Liouville%E2%80%93Neumann_series/faVVRLCV">Liouville–Neumann series</a>) was by <a href="/facts/Ivar_Fredholm/x4YvpYLv">Ivar Fredholm</a>, in a landmark 1903 paper in Acta Mathematica that helped establish modern <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>.
The name resolvent was given by <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a>.

<h2 id="resolvent-identity">Resolvent identity</h2>
For all z, w in ρ(A), the <a href="/facts/Resolvent_set/f4ItQgNP">resolvent set</a> of an operator A, we have that the first resolvent identity (also called Hilbert's identity) holds:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

R
        (
        z
        ;
        A
        )
        −
        R
        (
        w
        ;
        A
        )
        =
        (
        z
        −
        w
        )
        R
        (
        z
        ;
        A
        )
        R
        (
        w
        ;
        A
        )
        
        .
      
    
    {\displaystyle R(z;A)-R(w;A)=(z-w)R(z;A)R(w;A)\,.}

(Note that <a href="/facts/Dunford_and_Schwartz/RRWBcSUL">Dunford and Schwartz</a>, cited, define the resolvent as (zI −A)−1, instead, so that the formula above differs in sign from theirs.)
The second resolvent identity is a generalization of the first resolvent identity, above, useful for comparing the resolvents of two distinct operators. Given operators A and B, both defined on the same linear space, and z in ρ(A) ∩ ρ(B) the following identity holds,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

R
        (
        z
        ;
        A
        )
        −
        R
        (
        z
        ;
        B
        )
        =
        R
        (
        z
        ;
        A
        )
        (
        B
        −
        A
        )
        R
        (
        z
        ;
        B
        )
        
        .
      
    
    {\displaystyle R(z;A)-R(z;B)=R(z;A)(B-A)R(z;B)\,.}

A one-line proof goes as follows:

(
        A
        −
        z
        I
        
          )
          
            −
            1
          
        
        −
        (
        B
        −
        z
        I
        
          )
          
            −
            1
          
        
        =
        (
        A
        −
        z
        I
        
          )
          
            −
            1
          
        
        (
        (
        B
        −
        z
        I
        )
        −
        (
        A
        −
        z
        I
        )
        )
        (
        B
        −
        z
        I
        
          )
          
            −
            1
          
        
        =
        (
        A
        −
        z
        I
        
          )
          
            −
            1
          
        
        (
        B
        −
        A
        )
        (
        B
        −
        z
        I
        
          )
          
            −
            1
          
        
        
        .
      
    
    {\displaystyle (A-zI)^{-1}-(B-zI)^{-1}=(A-zI)^{-1}((B-zI)-(A-zI))(B-zI)^{-1}=(A-zI)^{-1}(B-A)(B-zI)^{-1}\,.}

<h2 id="compact-resolvent">Compact resolvent</h2>
When studying a closed <a href="/facts/Unbounded_operator/5u2njqHt">unbounded operator</a> A: H → H on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> H, if there exists 
 
 
 
 z
 ∈
 ρ
 (
 A
 )
 
 
 {\displaystyle z\in \rho (A)}
 
 such that 
 
 
 
 R
 (
 z
 ;
 A
 )
 
 
 {\displaystyle R(z;A)}
 
 is a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a>, we say that A has compact resolvent. The spectrum 
 
 
 
 σ
 (
 A
 )
 
 
 {\displaystyle \sigma (A)}
 
 of such A is a discrete subset of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
. If furthermore A is <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a>, then 
 
 
 
 σ
 (
 A
 )
 ⊂
 
 R
 
 
 
 {\displaystyle \sigma (A)\subset \mathbb {R} }
 
 and there exists an orthonormal basis 
 
 
 
 {
 
 v
 
 i
 
 
 
 }
 
 i
 ∈
 
 N
 
 
 
 
 
 {\displaystyle \{v_{i}\}_{i\in \mathbb {N} }}
 
 of eigenvectors of A with eigenvalues 
 
 
 
 {
 
 λ
 
 i
 
 
 
 }
 
 i
 ∈
 
 N
 
 
 
 
 
 {\displaystyle \{\lambda _{i}\}_{i\in \mathbb {N} }}
 
 respectively. Also, 
 
 
 
 {
 
 λ
 
 i
 
 
 }
 
 
 {\displaystyle \{\lambda _{i}\}}
 
 has no finite <a href="/facts/Accumulation_point/95MykoGU">accumulation point</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Resolvent_set/f4ItQgNP">Resolvent set</a></li>
<li><a href="/facts/Stone%27s_theorem_on_one-parameter_unitary_groups/1quJvzU7">Stone's theorem on one-parameter unitary groups</a></li>
<li><a href="/facts/Holomorphic_functional_calculus/xtWI9f4V">Holomorphic functional calculus</a></li>
<li><a href="/facts/Spectral_theory/8sAEEIMM">Spectral theory</a></li>
<li><a href="/facts/Compact_operator/KNkcK4mE">Compact operator</a></li>
<li><a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a></li>
<li><a href="/facts/Fredholm_theory/q1Kkxd1I">Fredholm theory</a></li>
<li><a href="/facts/Liouville%E2%80%93Neumann_series/faVVRLCV">Liouville–Neumann series</a></li>
<li><a href="/facts/Decomposition_of_spectrum_(functional_analysis)/6jQ7KdsE">Decomposition of spectrum (functional analysis)</a></li>
<li><a href="/facts/Limiting_absorption_principle/qPuNt2jD">Limiting absorption principle</a></li></ul>

<ul><li><a href="/facts/Nelson_Dunford/Q72PJMF7">Dunford, Nelson</a>; <a href="/facts/Jacob_T._Schwartz/UAvyPwCr">Schwartz, Jacob T.</a> (1988), Linear Operators, Part I General Theory, Hoboken, NJ: Wiley-Interscience, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-60848-3</li>
<li><a href="/facts/Erik_Ivar_Fredholm/x4YvpYLv">Fredholm, Erik I.</a> (1903), <a href="https://zenodo.org/record/1925972/files/article.pdf">"Sur une classe d'equations fonctionnelles"</a> (PDF), Acta Mathematica, 27: 365–390, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02421317">10.1007/bf02421317</a></li>
<li><a href="/facts/Einar_Hille/56qNTKUA">Hille, Einar</a>; <a href="/facts/Ralph_S._Phillips/JW4EJ7v3">Phillips, Ralph S.</a> (1957), Functional Analysis and Semi-groups, Providence: American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1031-6 {{citation}}: ISBN / Date incompatibility (help).</li>
<li><a href="/facts/Tosio_Kato/k4coaLQh">Kato, Tosio</a> (1980), Perturbation Theory for Linear Operators (2nd ed.), New York, NY: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-07558-5.</li>
<li><a href="/facts/Michael_E._Taylor/gfaKgoE5">Taylor, Michael E.</a> (1996), Partial Differential Equations I, New York, NY: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 7-5062-4252-4</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Taylor, section 9 of Appendix A. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hille and Phillips, Theorem 11.4.1, p. 341 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dunford and Schwartz, Vol I, Lemma 6, p. 568. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hille and Phillips, Theorem 4.8.2, p. 126 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Taylor, p. 515. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Resolvent formalism open-in-new

Resolvent formalism