# Carleman operator

A Carleman operator on the space $L _ {2} ( X, \mu )$ is an 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; Series expansion), which can be used in the study of linear elliptic operators, see [a1]. A general reference for Carleman operators on $L _ {2}$- spaces is [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 [a3]) and by considering Bochner integrals (cf. Bochner integral) and abstract Banach spaces one is lead to the so-called Carleman and Korotkov operators on a Banach space ([a4]).