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 ![]() |
[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 |
Integral representations of linear operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_representations_of_linear_operators&oldid=19185