View source for Spectrum of an operator

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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).

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.

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

<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>

<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>

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.

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" />:

<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>

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.

The spectra of operator functions are defined by the formula

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(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" />.

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]].

====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>



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====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>