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]).

References

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