Decomposition of spectrum (functional analysis)

<p>The <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> of a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> 
  
    
      
        T
      
    
    {\displaystyle T}
  
 that operates on a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a fundamental concept of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>. The spectrum consists of all <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 such that the operator 
  
    
      
        T
        −
        λ
      
    
    {\displaystyle T-\lambda }
  
 does not have a bounded <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a> on 
  
    
      
        X
      
    
    {\displaystyle X}
  
.  The spectrum has a standard decomposition into three parts: 
</p>
<ul><li>a point spectrum, consisting of the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of 
  
    
      
        T
      
    
    {\displaystyle T}
  
;</li>
<li>a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of 
  
    
      
        T
        −
        λ
      
    
    {\displaystyle T-\lambda }
  
 a <a href="/facts/Proper_subset/SJGERmbA">proper</a> <a href="/facts/Dense_subset/nPT9Yxao">dense subset</a> of the space;</li>
<li>a residual spectrum, consisting of all other scalars in the spectrum.</li></ul>
<p>This decomposition is relevant to the study of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>, and has applications to many branches of science and engineering. A well-known example from <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> is the explanation for the <a href="/facts/Discrete_spectrum_(physics)/L4UYDH1x">discrete spectral lines</a> and the continuous band in the light emitted by <a href="/facts/Excited_state/czcLYNnr">excited</a> atoms of <a href="/facts/Hydrogen/BoYHjl0J">hydrogen</a>.
</p>

Decomposition of spectrum (functional analysis) open-in-new

Decomposition of spectrum (functional analysis)