View source for Minimax property

''of eigen values''

A special type of relationship connecting the eigen values of a completely-continuous [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639601.png" /> (cf. also [[Completely-continuous operator|Completely-continuous operator]]) with the maximum and minimum values of the associated quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639602.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639603.png" /> be a completely-continuous self-adjoint operator on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639604.png" />. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639605.png" /> consists of a finite or countable set of real eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639606.png" /> having unique limit point zero. The root subspaces corresponding to the non-zero eigen values consist of eigen vectors and are finite dimensional; the eigen subspaces associated with distinct eigen values are mutually orthogonal; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639607.png" /> has a complete system of eigen vectors. The spectral decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639608.png" /> (cf. [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]) has the form: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396010.png" /> are the distinct eigen values, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396011.png" /> are the projection operators onto the corresponding eigen spaces, and the series converges in the operator norm. The norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396012.png" /> coincides with the maximum modulus of the eigen values and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396013.png" />; the maximum is attained at the corresponding eigen vector.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396014.png" /> be the positive eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396015.png" />, where each eigen value is repeated as often as its multiplicity. Then

<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/m/m063/m063960/m06396016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396017.png" /> are arbitrary non-zero vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396018.png" />. Similar relations hold for the negative eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396019.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/m/m063/m063960/m06396020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Relations (1) and (2) are applied for finding the eigen values of integral operators with a symmetric kernel. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396022.png" /> are completely-continuous self-adjoint operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396023.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396024.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396026.png" /> the sequences of their positive eigen values, listed in decreasing order, where each value is repeated as often as its multiplicity, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396027.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Wiley  (1988)</TD></TR></table>