Resolvent formalism

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the resolvent formalism is a technique for applying concepts from <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> to the study of the <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> of <a href="/facts/Operator_(mathematics)/1VWgEdjS">operators</a> on <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> and more general spaces. Formal justification for the manipulations can be found in the framework of <a href="/facts/Holomorphic_functional_calculus/xtWI9f4V">holomorphic functional calculus</a>.
The resolvent captures the spectral properties of an operator in the analytic structure of the <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a>. Given an operator A, the resolvent may be defined as

R
        (
        z
        ;
        A
        )
        =
        (
        A
        −
        z
        I
        
          )
          
            −
            1
          
        
         
        .
      
    
    {\displaystyle R(z;A)=(A-zI)^{-1}~.}

Among other uses, the resolvent may be used to solve the inhomogeneous <a href="/facts/Fredholm_integral_equation/euHGJbQo">Fredholm integral equations</a>; a commonly used approach is a series solution, the <a href="/facts/Liouville%E2%80%93Neumann_series/faVVRLCV">Liouville–Neumann series</a>.
The resolvent of A can be used to directly obtain information about the <a href="/facts/Spectral_theorem/Vqc04B21">spectral decomposition</a>
of A. For example, suppose λ is an isolated <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> in the 
<a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> of A. That is, suppose there exists a simple closed curve 
 
 
 
 
 C
 
 λ
 
 
 
 
 {\displaystyle C_{\lambda }}
 
 
in the complex plane that separates λ from the rest of the spectrum of A.
Then the <a href="/facts/Residue_(complex_analysis)/GllQJk32">residue</a>

−
        
          
            1
            
              2
              π
              i
            
          
        
        
          ∮
          
            
              C
              
                λ
              
            
          
        
        (
        A
        −
        z
        I
        
          )
          
            −
            1
          
        
         
        d
        z
      
    
    {\displaystyle -{\frac {1}{2\pi i}}\oint _{C_{\lambda }}(A-zI)^{-1}~dz}

defines a <a href="/facts/Projection_operator/sElXGkxD">projection operator</a> onto the λ <a href="/facts/Eigenspace/8TjEoT8u">eigenspace</a> of A. 
The <a href="/facts/Hille%E2%80%93Yosida_theorem/s1qSoQuq">Hille–Yosida theorem</a> relates the resolvent through a <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> to an integral over the one-parameter <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of transformations generated by A. Thus, for example, if A is a <a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">skew-Hermitian matrix</a>, then U(t) = exp(tA) is a one-parameter group of unitary operators. Whenever 
 
 
 
 
 |
 
 z
 
 |
 
 >
 ‖
 A
 ‖
 
 
 {\displaystyle |z|>\|A\|}
 
, the resolvent of A at z can be expressed as the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a>

R
        (
        z
        ;
        A
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        
          e
          
            −
            z
            t
          
        
        U
        (
        t
        )
         
        d
        t
        ,
      
    
    {\displaystyle R(z;A)=\int _{0}^{\infty }e^{-zt}U(t)~dt,}

where the integral is taken along the ray 
 
 
 
 arg
 ⁡
 t
 =
 −
 arg
 ⁡
 λ
 
 
 {\displaystyle \arg t=-\arg \lambda }
 
.

Resolvent formalism open-in-new

Resolvent formalism