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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866102.png" /> of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866103.png" /> for which the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866104.png" /> does not have an everywhere-defined bounded inverse. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866105.png" /> is a [[Linear operator|linear operator]] on a complex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866107.png" /> is the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s0866109.png" /> is not closed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661011.png" />, and therefore one usually considers spectra of closed operators (the spectrum of the closure of an operator for operators admitting a closure is sometimes called the closure spectrum).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661012.png" /> is either non-injective or non-surjective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661013.png" />. In the first case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661014.png" /> is called an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661015.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661016.png" /> of eigenvalues is called the point spectrum. In the second case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661017.png" /> is called a point of the continuous spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661018.png" /> or the residual spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661019.png" />, depending on whether the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661020.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661021.png" /> or not.
+
'' $  A $''
  
There are also other classifications of the points of a spectrum. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661023.png" /> consists of approximate eigenvalues (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661024.png" /> if there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661027.png" />), and
+
The set  $  \sigma ( A) $
 +
of complex numbers  $  \lambda \in \mathbf C $
 +
for which the operator  $  A- \lambda I $
 +
does not have an everywhere-defined bounded inverse. Here,  $  A $
 +
is a [[Linear operator|linear operator]] on a complex Banach space  $  X $
 +
and  $  I $
 +
is the identity operator on  $  X $.  
 +
If  $  A $
 +
is not closed on  $  X $,  
 +
then  $  \sigma ( A) = \mathbf C $,  
 +
and therefore one usually considers spectra of closed operators (the spectrum of the closure of an operator for operators admitting a closure is sometimes called the closure spectrum).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661028.png" /></td> </tr></table>
+
If  $  A- \lambda I $
 +
is either non-injective or non-surjective, then  $  \lambda \in \sigma ( A) $.
 +
In the first case  $  \lambda $
 +
is called an eigenvalue of  $  A $;  
 +
the set  $  \sigma _ {p} ( A) $
 +
of eigenvalues is called the point spectrum. In the second case  $  \lambda $
 +
is called a point of the continuous spectrum  $  \sigma _ {c} ( A) $
 +
or the residual spectrum  $  \sigma _ {r} ( A) $,
 +
depending on whether the subspace  $  ( A- \lambda I) X $
 +
is dense in  $  X $
 +
or not.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661029.png" /></td> </tr></table>
+
There are also other classifications of the points of a spectrum. For example,  $  \sigma ( A)= \sigma _ {a} ( A) \cup \sigma _ {d} ( A) $,
 +
where  $  \sigma _ {a} ( A) $
 +
consists of approximate eigenvalues ( $  \lambda \in \sigma _ {a} ( A) $
 +
if there are  $  \{ x _ {n} \} \subset  X $
 +
with  $  \| x _ {n} \| = 1 $
 +
such that  $  \| ( A- \lambda I) x _ {n} \| \rightarrow 0 $),
 +
and
  
Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661030.png" />, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661031.png" />. In perturbation theory, use is made of the limit spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661032.png" />, which consists of the limit points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661033.png" /> and the isolated eigenvalues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.
+
$$
 +
\sigma _ {d} ( A) =
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661034.png" /> is a [[Bounded operator|bounded operator]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661035.png" /> is compact and non-empty (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661036.png" /> coincides with the spectrum of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661037.png" /> of the Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661038.png" />, cf. [[Spectrum of an element|Spectrum of an element]]); in general one can only say that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661039.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661040.png" />. On the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661041.png" /> one can define the analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661042.png" />-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661043.png" />, called the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661045.png" /> is called the resolvent set). With the help of resolvents a functional calculus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661046.png" /> is built on functions analytic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661047.png" />:
+
$$
 +
= \
 +
\{ {\lambda \in \mathbf C } : { \mathop{\rm Ker} ( A- \lambda I) = 0 ,
 +
\overline{ {( A- \lambda I ) X }}\; = ( A - \lambda I ) X \neq X } \} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661048.png" /></td> </tr></table>
+
Note that  $  \sigma _ {d} ( A) \subset  \sigma _ {r} ( A) $,
 +
and so  $  \sigma _ {p} ( A) \cup \sigma _ {c} ( A) \subset  \sigma _ {a} ( A) $.  
 +
In perturbation theory, use is made of the limit spectrum  $  \sigma _ {\lim\limits} ( A) $,
 +
which consists of the limit points of  $  \sigma ( A) $
 +
and the isolated eigenvalues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661049.png" /> is a contour enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661050.png" /> (the unboundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661051.png" /> imposes restrictions on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661052.png" />). Further conditions on the geometry of the spectrum and on the asymptotics of the resolvent enables one to extend this calculus.
+
If  $  A $
 +
is a [[Bounded operator|bounded operator]], then  $  \sigma ( A) $
 +
is compact and non-empty (in this case  $  \sigma ( A) $
 +
coincides with the spectrum of the element  $  A $
 +
of the Banach algebra  $  B( X) $,
 +
cf. [[Spectrum of an element|Spectrum of an element]]); in general one can only say that  $  \sigma ( A) $
 +
is closed in  $  \mathbf C $.
 +
On the set  $  \rho ( A) = \mathbf C \setminus  \sigma ( A) $
 +
one can define the analytic  $  B( X) $-
 +
valued function  $  R _ {A} ( \lambda ) = ( A- \lambda I )  ^ {-} 1 $,
 +
called the resolvent of  $  A $(
 +
$  \rho ( A) $
 +
is called the resolvent set). With the help of resolvents a functional calculus for  $  A $
 +
is built on functions analytic in a neighbourhood of  $  \sigma ( A) $:
 +
 
 +
$$
 +
f( A)  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma  f( \lambda ) R _ {A} ( \lambda )  d \lambda ,
 +
$$
 +
 
 +
where  $  \Gamma $
 +
is a contour enclosing $  \sigma ( A) $(
 +
the unboundedness of $  A $
 +
imposes restrictions on the choice of $  \Gamma $).  
 +
Further conditions on the geometry of the spectrum and on the asymptotics of the resolvent enables one to extend this calculus.
  
 
The spectra of operator functions are defined by the formula
 
The spectra of operator functions are defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661053.png" /></td> </tr></table>
+
$$
 +
\sigma ( f( A))  = \{ {f( \lambda ) } : {\lambda \in \sigma ( A) } \}
 +
$$
  
(the spectral mapping theorem). The spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661054.png" /> of the adjoint operator coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661055.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661056.png" /> is bounded; in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661057.png" />.
+
(the spectral mapping theorem). The spectrum $  \sigma ( A  ^ {*} ) $
 +
of the adjoint operator coincides with $  \sigma ( A) $
 +
when $  A $
 +
is bounded; in general, $  \sigma ( A  ^ {*} ) \subset  \sigma ( A) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661060.png" /> decomposes into the direct sum of subspaces invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661061.png" />, on each of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086610/s08661062.png" /> induces an operator with one-point spectrum. [[Spectral theory|Spectral theory]] of operators is concerned with finding infinite-dimensional analogues for this decomposition. See also [[Spectral analysis|Spectral analysis]]; [[Spectral synthesis|Spectral synthesis]]; [[Spectral operator|Spectral operator]]; [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]].
+
If $  \mathop{\rm dim}  X < \infty $,  
 +
then $  \sigma ( A)= \sigma _ {p} ( A) $,  
 +
and $  X $
 +
decomposes into the direct sum of subspaces invariant under $  A $,  
 +
on each of which $  A $
 +
induces an operator with one-point spectrum. [[Spectral theory|Spectral theory]] of operators is concerned with finding infinite-dimensional analogues for this decomposition. See also [[Spectral analysis|Spectral analysis]]; [[Spectral synthesis|Spectral synthesis]]; [[Spectral operator|Spectral operator]]; [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1980)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>

Latest revision as of 08:22, 6 June 2020


$ A $

The set $ \sigma ( A) $ of complex numbers $ \lambda \in \mathbf C $ for which the operator $ A- \lambda I $ does not have an everywhere-defined bounded inverse. Here, $ A $ is a linear operator on a complex Banach space $ X $ and $ I $ is the identity operator on $ X $. If $ A $ is not closed on $ X $, then $ \sigma ( A) = \mathbf C $, and therefore one usually considers spectra of closed operators (the spectrum of the closure of an operator for operators admitting a closure is sometimes called the closure spectrum).

If $ A- \lambda I $ is either non-injective or non-surjective, then $ \lambda \in \sigma ( A) $. In the first case $ \lambda $ is called an eigenvalue of $ A $; the set $ \sigma _ {p} ( A) $ of eigenvalues is called the point spectrum. In the second case $ \lambda $ is called a point of the continuous spectrum $ \sigma _ {c} ( A) $ or the residual spectrum $ \sigma _ {r} ( A) $, depending on whether the subspace $ ( A- \lambda I) X $ is dense in $ X $ or not.

There are also other classifications of the points of a spectrum. For example, $ \sigma ( A)= \sigma _ {a} ( A) \cup \sigma _ {d} ( A) $, where $ \sigma _ {a} ( A) $ consists of approximate eigenvalues ( $ \lambda \in \sigma _ {a} ( A) $ if there are $ \{ x _ {n} \} \subset X $ with $ \| x _ {n} \| = 1 $ such that $ \| ( A- \lambda I) x _ {n} \| \rightarrow 0 $), and

$$ \sigma _ {d} ( A) = $$

$$ = \ \{ {\lambda \in \mathbf C } : { \mathop{\rm Ker} ( A- \lambda I) = 0 , \overline{ {( A- \lambda I ) X }}\; = ( A - \lambda I ) X \neq X } \} . $$

Note that $ \sigma _ {d} ( A) \subset \sigma _ {r} ( A) $, and so $ \sigma _ {p} ( A) \cup \sigma _ {c} ( A) \subset \sigma _ {a} ( A) $. In perturbation theory, use is made of the limit spectrum $ \sigma _ {\lim\limits} ( A) $, which consists of the limit points of $ \sigma ( A) $ and the isolated eigenvalues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.

If $ A $ is a bounded operator, then $ \sigma ( A) $ is compact and non-empty (in this case $ \sigma ( A) $ coincides with the spectrum of the element $ A $ of the Banach algebra $ B( X) $, cf. Spectrum of an element); in general one can only say that $ \sigma ( A) $ is closed in $ \mathbf C $. On the set $ \rho ( A) = \mathbf C \setminus \sigma ( A) $ one can define the analytic $ B( X) $- valued function $ R _ {A} ( \lambda ) = ( A- \lambda I ) ^ {-} 1 $, called the resolvent of $ A $( $ \rho ( A) $ is called the resolvent set). With the help of resolvents a functional calculus for $ A $ is built on functions analytic in a neighbourhood of $ \sigma ( A) $:

$$ f( A) = \frac{1}{2 \pi i } \int\limits _ \Gamma f( \lambda ) R _ {A} ( \lambda ) d \lambda , $$

where $ \Gamma $ is a contour enclosing $ \sigma ( A) $( the unboundedness of $ A $ imposes restrictions on the choice of $ \Gamma $). Further conditions on the geometry of the spectrum and on the asymptotics of the resolvent enables one to extend this calculus.

The spectra of operator functions are defined by the formula

$$ \sigma ( f( A)) = \{ {f( \lambda ) } : {\lambda \in \sigma ( A) } \} $$

(the spectral mapping theorem). The spectrum $ \sigma ( A ^ {*} ) $ of the adjoint operator coincides with $ \sigma ( A) $ when $ A $ is bounded; in general, $ \sigma ( A ^ {*} ) \subset \sigma ( A) $.

If $ \mathop{\rm dim} X < \infty $, then $ \sigma ( A)= \sigma _ {p} ( A) $, and $ X $ decomposes into the direct sum of subspaces invariant under $ A $, on each of which $ A $ induces an operator with one-point spectrum. Spectral theory of operators is concerned with finding infinite-dimensional analogues for this decomposition. See also Spectral analysis; Spectral synthesis; Spectral operator; Spectral decomposition of a linear operator.

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[2] T. Kato, "Perturbation theory for linear operators" , Springer (1980)

Comments

References

[a1] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
[a2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
How to Cite This Entry:
Spectrum of an operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_an_operator&oldid=11393
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article