View source for Hilbert-Schmidt integral operator

A bounded linear [[Integral operator|integral operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473301.png" /> acting from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473302.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473303.png" /> and representable in the form

<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/h/h047/h047330/h0473304.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473305.png" /> is the kernel of the operator (cf. [[Kernel of an integral operator|Kernel of an integral operator]], [[#References|[1]]]).

D. Hilbert and E. Schmidt in 1907 were the first to study operators of this kind. A Hilbert–Schmidt integral operator is a [[Completely-continuous operator|completely-continuous operator]] [[#References|[2]]]. Its adjoint is also a Hilbert–Schmidt integral operator, with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473306.png" /> [[#References|[3]]]. A Hilbert–Schmidt integral operator is a [[Self-adjoint operator|self-adjoint operator]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473307.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473308.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h0473309.png" />). For a self-adjoint Hilbert–Schmidt integral operator and for its kernel the following expansions are valid:

<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/h/h047/h047330/h04733010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/h/h047/h047330/h04733011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733012.png" /> is the orthonormal system of eigen functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733013.png" /> corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733014.png" />. The series (1) converges with respect to the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733015.png" />, while the series (2) converges with respect to the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733016.png" />, [[#References|[4]]]. Under the conditions of the [[Mercer theorem|Mercer theorem]] the series (2) converges absolutely and uniformly [[#References|[5]]].

If

<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/h/h047/h047330/h04733017.png" /></td> </tr></table>

then the series (1) converges absolutely and uniformly, [[#References|[4]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733018.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733019.png" />-finite measure, then the linear operator

<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/h/h047/h047330/h04733020.png" /></td> </tr></table>

is a Hilbert–Schmidt integral operator if and only if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733021.png" /> such that the inequality

<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/h/h047/h047330/h04733022.png" /></td> </tr></table>

is valid for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733023.png" /> (with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733024.png" />) [[#References|[7]]]. Thus, the Hilbert–Schmidt integral operators form a two-sided ideal in the Banach algebra of all bounded linear operators acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733025.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047330/h04733026.png" />.

Hilbert–Schmidt integral operators play an important role in the theory of integral equations and in the theory of boundary value problems [[#References|[8]]], [[#References|[9]]], because the operators which appear in many problems of mathematical physics are either themselves Hilbert–Schmidt integral operators or else their iteration to a certain order is such an operator. A natural generalization of a Hilbert–Schmidt integral operator is a [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.H. Stone,   "Linear transformations in Hilbert space and their applications to analysis" , Amer. Math. Soc.  (1932)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Riesz,   B. Szökefalvi-Nagy,   "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Dieudonné,   "Foundations of modern analysis" , Acad. Press  (1961)  (Translated from French)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.V. Kantorovich,   B.Z. Vulikh,   A.G. Pinsker,   "Functional analysis in semi-ordered spaces" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Weidmann,   "Carleman operatoren"  ''Manuscripta Math.'' , '''2''' :  1  (1970)  pp. 1–38</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  K. Moren,   "Methods of Hilbert spaces" , PWN  (1967)  (Translated from Polish)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR></table>