Integral representations of linear operators

Let $( X, \Lambda, \mu )$ and $( Y, \Sigma, \nu )$ be $\sigma$-finite measure spaces (cf. Measure space) and let $L _ {0} ( X, \mu )$ and $L _ {0} ( Y, \nu )$ be the spaces of the complex-valued $\mu$-measurable functions on $X$ and the complex-valued $\nu$-measurable functions on $Y$, respectively. A linear subspace $E = E ( X, \mu )$ of $L _ {0} ( X, \mu )$ is called an ideal space, or a solid linear subspace, of $L _ {0}$ if $f \in L _ {0}$, $g \in E$ and $| f | \leq | g |$, $\mu$-a.e., imply $f \in E$. The classical $L _ {p}$-spaces ( $1 \leq p \leq \infty$), the Orlicz spaces and, more generally, Banach function spaces (cf. also Orlicz space; Banach space) are typical examples of normed ideal spaces.

If $E$, $F$ are ideal spaces contained in $L _ {0} ( Y, \nu )$ and $L _ {0} ( X, \mu )$, respectively, then $T \in {\mathcal L} ( E,F )$, the linear space of all linear operators from $E$ into $F$, is called an integral operator, kernel operator, if there exists a $( \mu \times \nu )$-measurable function $T = T ( x,y )$, $( x,y ) \in X \times Y$, such that for all $f \in E$ and $\mu$-a.e. with respect to $x$, $( Tf ) ( x ) = \int _ {Y} {T ( x,y ) f ( y ) } {d \nu ( y ) }$.

Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which $T \in {\mathcal L} ( E,F )$ are integral operators? J. von Neumann [a5] was the first to show that for the ideal spaces $E = F = L _ {2} ( [ 0,1 ] )$ the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator $T \in {\mathcal L} ( L _ {2} , L _ {2} )$ is unitarily equivalent (cf. also Unitarily-equivalent operators) to an integral operator if and only if $0$ is an element of the limit spectrum of $T$.

$T \in {\mathcal L} ( E,F )$ is called a positive linear operator if for all $0 \leq f \in E$ one has $T f \geq 0$ ($\mu$-a.e.). An integral operator $T$ with kernel $T ( x,y )$ ($( x,y ) \in X \times Y$) is positive if and only if $T ( x,y ) \geq 0$, $( \mu \times \nu )$-a.e.; $T \in {\mathcal L} ( E,F )$ is called regular if $T$ maps order-bounded sets into order-bounded sets, i.e., for all $f \in E$ there exists a $g \in F$ such that for all $h \in E$ satisfying $| h | \leq | f |$, one has $| {Th } | \leq g$; $T \in {\mathcal L} ( E,F )$ is ordered bounded if and only if $T$ can be written as the difference of two positive linear operators if and only if its modulus $| T |$, where for all $0 \leq f \in E$, $| T | ( f ) = \sup \{ {| {Tg } | } : {| g | \leq f } \}$, is a positive linear operator mapping $E$ into $F$.

The following theorem holds: An integral operator $T \in {\mathcal L} ( E,F )$ is regular if and only if its modulus $| T |$ is a positive linear operator mapping $E$ into $F$. In that case, the kernel of $| T |$ is given by the modulus $| {T ( x,y ) } |$ ($( x,y ) \in X \times Y$) of the kernel of $T$.

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: $T \in {\mathcal L} ( E,F )$ is a linear integral operator if and only if $0 \leq f _ {n} \leq f \in E$ ($n = 1,2, \dots$) and $f _ {n} \rightarrow 0$ in $\nu$-measure as $n \rightarrow \infty$ imply $Tf _ {n} \rightarrow 0$ ($\mu$-a.e.) as $n \rightarrow \infty$.

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 $K$-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