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A Carleman operator on the space  $  L _ {2} ( X, \mu ) $
is an [[Integral operator|integral operator]]  $  T $, 
i.e.,  $  Tf ( x ) = \int {T ( x,y ) f ( y ) }  {d \mu ( y ) } $
a.e. for  $  f \in L _ {2} ( X, \mu ) $, 
such that  $  \| {T ( x, \cdot ) } \| _ {2} < \infty $
a.e. on  $  X $. 
Self-adjoint Carleman operators have generalized eigenfunction expansions (cf. [[Eigen function|Eigen function]]; [[Series expansion|Series expansion]]), which can be used in the study of linear elliptic operators, see [[#References|[a1]]]. A general reference for Carleman operators on  $  L _ {2} $-
spaces is [[#References|[a2]]]. The notion of a Carleman operator has been extended in many directions. By replacing  $  L _ {2} $
by an arbitrary Banach function space one obtains the so-called generalized Carleman operators (see [[#References|[a3]]]) and by considering Bochner integrals (cf. [[Bochner integral|Bochner integral]]) and abstract Banach spaces one is lead to the so-called Carleman and Korotkov operators on a [[Banach space|Banach space]] ([[#References|[a4]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Maurin,   "Methods of Hilbert spaces" , PWN  (1967)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   V.S. Sunder,   "Bounded integral operators on $L^2$-spaces" , ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' , '''96''' , Springer  (1978)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Schep,   "Generalized Carleman operators"  ''Indagationes Mathematicae'' , '''42'''  (1980)  pp. 49–59</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Gretsky,   J.J. Uhl,   "Carleman and Korotkov operators on Banach spaces"  ''Acta Sci. Math'' , '''43'''  (1981)  pp. 111–119</TD></TR>
</table>