Spectrum (functional analysis)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the spectrum of a <a href="/facts/Bounded_operator/LYmDTIqR">bounded linear operator</a> (or, more generally, an <a href="/facts/Unbounded_operator/5u2njqHt">unbounded linear operator</a>) is a generalisation of the set of <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>. Specifically, a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is said to be in the spectrum of a bounded linear operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 if 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}

<ul><li>either has no set-theoretic <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a>;</li>
<li>or the set-theoretic inverse is either unbounded or defined on a non-dense subset.</li></ul>
Here, 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is the <a href="/facts/Identity_function/9AtsP0LC">identity operator</a>.
By the <a href="/facts/Closed_graph_theorem/CK2ra00W">closed graph theorem</a>, 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is in the spectrum if and only if the bounded operator 
 
 
 
 T
 −
 λ
 I
 :
 V
 →
 V
 
 
 {\displaystyle T-\lambda I:V\to V}
 
 is non-bijective on 
 
 
 
 V
 
 
 {\displaystyle V}
 
.
The study of spectra and related properties is known as <a href="/facts/Spectral_theory/8sAEEIMM">spectral theory</a>, which has numerous applications, most notably the <a href="/facts/Mathematical_formulation_of_quantum_mechanics/fRLw4sFU">mathematical formulation of quantum mechanics</a>.
The spectrum of an operator on a <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the <a href="/facts/Unilateral_shift/6c6lc0ks">right shift</a> operator R on the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <a href="/facts/Lp_space/gbad0Rv1">ℓ2</a>,

(
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        )
        ↦
        (
        0
        ,
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        )
        .
      
    
    {\displaystyle (x_{1},x_{2},\dots )\mapsto (0,x_{1},x_{2},\dots ).}

This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in <a href="/facts/Lp_space/gbad0Rv1">ℓ2</a>. In fact every bounded linear operator on a <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> must have a non-empty spectrum.
The notion of spectrum extends to <a href="/facts/Unbounded_operator/5u2njqHt">unbounded</a> (i.e. not necessarily bounded) operators. A <a href="/facts/Complex_number/2w7mqI7e">complex number</a> λ is said to be in the spectrum of an unbounded operator 
 
 
 
 T
 :
 
 X
 →
 X
 
 
 {\displaystyle T:\,X\to X}
 
 defined on domain 
 
 
 
 D
 (
 T
 )
 ⊆
 X
 
 
 {\displaystyle D(T)\subseteq X}
 
 if there is no bounded inverse 
 
 
 
 (
 T
 −
 λ
 I
 
 )
 
 −
 1
 
 
 :
 
 X
 →
 D
 (
 T
 )
 
 
 {\displaystyle (T-\lambda I)^{-1}:\,X\to D(T)}
 
 defined on the whole of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 If T is <a href="/facts/Closed_operator/5u2njqHt">closed</a> (which includes the case when T is bounded), boundedness of 
 
 
 
 (
 T
 −
 λ
 I
 
 )
 
 −
 1
 
 
 
 
 {\displaystyle (T-\lambda I)^{-1}}
 
 follows automatically from its existence. 
The space of bounded linear operators B(X) on a Banach space X is an example of a <a href="/facts/Unital_algebra/8T8ulWJb">unital</a> <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a>. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

Spectrum (functional analysis) open-in-new

Spectrum (functional analysis)