Integral representations of linear operators

Let and be -finite measure spaces (cf. Measure space) and let and be the spaces of the complex-valued -measurable functions on and the complex-valued -measurable functions on , respectively. A linear subspace of is called an ideal space, or a solid linear subspace, of if , and , -a.e., imply . The classical -spaces (), the Orlicz spaces and, more generally, Banach function spaces (cf. also Orlicz space; Banach space) are typical examples of normed ideal spaces.

If , are ideal spaces contained in and , respectively, then , the linear space of all linear operators from into , is called an integral operator, kernel operator, if there exists a -measurable function , , such that for all and -a.e. with respect to , .

Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which are integral operators? J. von Neumann [a5] was the first to show that for the ideal spaces the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator is unitarily equivalent (cf. also Unitarily-equivalent operators) to an integral operator if and only if is an element of the limit spectrum of .

is called a positive linear operator if for all one has (-a.e.). An integral operator with kernel () is positive if and only if , -a.e.; is called regular if maps order-bounded sets into order-bounded sets, i.e., for all there exists a such that for all satisfying , one has ; is ordered bounded if and only if can be written as the difference of two positive linear operators if and only if its modulus , where for all , , is a positive linear operator mapping into .

The following theorem holds: An integral operator is regular if and only if its modulus is a positive linear operator mapping into . In that case, the kernel of is given by the modulus () of the kernel of .

An integral transform need not be regular, as is shown, for instance, by the Fourier transform and the Hilbert transform.

Integral operators can be characterized via a continuity property: is a linear integral operator if and only if () and in -measure as imply (-a.e.) as .

An earlier version of this theorem for bilinear forms is due to H. Nakano [a4]. For regular linear operators defined on KB-spaces (cf. also -space), the result appeared in a slightly different form in a paper by G.Ya. Lozonovskii [a3]. The present version is due to A.V. Bukhvalov [a1]. A pure measure-theoretic proof and related results were given by A. Schep [a6]. For details and further results see [a2].

References

[a1]	A.V. Bukhvalov, "A criterion for integral representability of linear operators" Funktsional. Anal. i Prilozhen. , 9 : 1 (1975) pp. 51 (In Russian)
[a2]	"Vector lattices and integral operators" S.S. Kutateladze (ed.) , Mathematics and its Applications , 358 , Kluwer Acad. Publ. (1996)
[a3]	G.Ya. Lozanovsky, "On almost integral operators in -spaces" Vestnik Leningrad Gos. Univ. , 7 (1966) pp. 35–44 (In Russian)
[a4]	H. Nakano, "Product spaces of semi-ordered linear spaces" J. Fac. Sci. Hokkaidô Univ. Ser. I , 12 : 3 (1953) pp. 163–210
[a5]	J. von Neumann, "Charakterisierung des Spektrums eines Integraloperators" , Actualités Sc. et Industr. , 229 , Hermann (1935)
[a6]	A.R. Schep, "Kernel operators" Proc. Kon. Nederl. Akad. Wetensch. , A 82 (1979) pp. 39–53