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A bounded linear [[Integral operator|integral operator]]  $  T $
acting from the space  $  L _ {2} ( X, \mu ) $
into  $  L _ {2} ( x, \mu ) $
and representable in the form

$$ 
( Tf  ) ( x)  =  \int\limits _ { X } K ( x, y) f ( y)  \mu ( dy),\ \ 
f \in L _ {2} ( X, \mu ),
$$

where  $  K ( \cdot , \cdot ) \in L _ {2} ( X \times X, \mu \times \mu ) $
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  $  \overline{ {K ( y, x ) }}\; $[[#References|[3]]]. A Hilbert–Schmidt integral operator is a [[Self-adjoint operator|self-adjoint operator]] if and only if  $  K ( x, y) = \overline{ {K ( y, x) }}\; $
for almost-all  $  ( x, y) \in X \times X $(
with respect to  $  ( \mu \times \mu ) $). 
For a self-adjoint Hilbert–Schmidt integral operator and for its kernel the following expansions are valid:

$$ \tag{1 }
( Tf  ) ( x)  = \ 
\sum _ { n } \lambda _ {n} ( f, \phi _ {n} ) \phi _ {n} ,\ \ 
f \in L _ {2} ( X, \mu ),
$$

$$ \tag{2 }
K ( x, y)  =  \sum _ { n } \lambda _ {n} \phi _ {n} ( x) \phi _ {n} ( y),
$$

where  $  \{ \phi _ {n} \} $
is the orthonormal system of eigen functions of  $  T $
corresponding to the eigen values  $  \lambda _ {n} \neq 0 $. 
The series (1) converges with respect to the norm of  $  L _ {2} ( X, \mu ) $, 
while the series (2) converges with respect to the norm of  $  L _ {2} ( X \times X, \mu \times \mu ) $, 
[[#References|[4]]]. Under the conditions of the [[Mercer theorem|Mercer theorem]] the series (2) converges absolutely and uniformly [[#References|[5]]].

If

$$ 
\int\limits _ { X }
| K ( x, y) |  ^ {2}  \mu ( dy)  \leq   C \ \ 
\textrm{ for }  \textrm{ all }  x \in X,
$$

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

If  $  \mu $
is a  $  \sigma $-
finite measure, then the linear operator

$$ 
T:  L _ {2} ( X, \mu )  \rightarrow  L _ {2} ( X, \mu )
$$

is a Hilbert–Schmidt integral operator if and only if there exists a function  $  M ( \cdot ) \in L _ {2} ( X, \mu ) $
such that the inequality

$$ 
| ( Tf  ) ( x) |  \leq   M ( x)  \| f \|
$$

is valid for almost-all  $  x \in X $(
with respect to the measure  $  \mu $) 
[[#References|[7]]]. Thus, the Hilbert–Schmidt integral operators form a two-sided ideal in the Banach algebra of all bounded linear operators acting from  $  L _ {2} ( X, \mu ) $
into  $  L _ {2} ( X, \mu ) $.

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>