View source for Integral representations of linear operators

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100802.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100803.png" />-finite measure spaces (cf. [[Measure space|Measure space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100805.png" /> be the spaces of the complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100806.png" />-measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100807.png" /> and the complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100808.png" />-measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100809.png" />, respectively. A linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008011.png" /> is called an ideal space, or a solid linear subspace, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008016.png" />-a.e., imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008017.png" />. The classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008019.png" />-spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008020.png" />), the Orlicz spaces and, more generally, Banach function spaces (cf. also [[Orlicz space|Orlicz space]]; [[Banach space|Banach space]]) are typical examples of normed ideal spaces.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008022.png" /> are ideal spaces contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008024.png" />, respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008025.png" />, the linear space of all linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008027.png" />, is called an integral operator, kernel operator, if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008028.png" />-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008030.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008032.png" />-a.e. with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008034.png" />.

Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008035.png" /> are integral operators? J. von Neumann [[#References|[a5]]] was the first to show that for the ideal spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008036.png" /> the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008037.png" /> is unitarily equivalent (cf. also [[Unitarily-equivalent operators|Unitarily-equivalent operators]]) to an integral operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008038.png" /> is an element of the limit spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008039.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008040.png" /> is called a positive linear operator if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008041.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008043.png" />-a.e.). An integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008044.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008046.png" />) is positive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008048.png" />-a.e.; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008049.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008050.png" /> maps order-bounded sets into order-bounded sets, i.e., for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008051.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008052.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008053.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008054.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008056.png" /> is ordered bounded if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008057.png" /> can be written as the difference of two positive linear operators if and only if its modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008058.png" />, where for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008060.png" />, is a positive linear operator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008061.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008062.png" />.

The following theorem holds: An integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008063.png" /> is regular if and only if its modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008064.png" /> is a positive linear operator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008065.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008066.png" />. In that case, the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008067.png" /> is given by the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008069.png" />) of the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008070.png" />.

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

Integral operators can be characterized via a continuity property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008071.png" /> is a linear integral operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008072.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008073.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008074.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008075.png" />-measure as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008076.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008077.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008078.png" />-a.e.) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008079.png" />.

An earlier version of this theorem for bilinear forms is due to H. Nakano [[#References|[a4]]]. For regular linear operators defined on KB-spaces (cf. also [[K-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008080.png" />-space]]), the result appeared in a slightly different form in a paper by G.Ya. Lozonovskii [[#References|[a3]]]. The present version is due to A.V. Bukhvalov [[#References|[a1]]]. A pure measure-theoretic proof and related results were given by A. Schep [[#References|[a6]]]. For details and further results see [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bukhvalov,   "A criterion for integral representability of linear operators"  ''Funktsional. Anal. i Prilozhen.'' , '''9''' :  1  (1975)  pp. 51  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  "Vector lattices and integral operators"  S.S. Kutateladze (ed.) , ''Mathematics and its Applications'' , '''358''' , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.Ya. Lozanovsky,   "On almost integral operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008081.png" />-spaces"  ''Vestnik Leningrad Gos. Univ.'' , '''7'''  (1966)  pp. 35–44  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Nakano,   "Product spaces of semi-ordered linear spaces"  ''J. Fac. Sci. Hokkaidô Univ. Ser. I'' , '''12''' :  3  (1953)  pp. 163–210</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. von Neumann,   "Charakterisierung des Spektrums eines Integraloperators" , ''Actualités Sc. et Industr.'' , '''229''' , Hermann  (1935)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.R. Schep,   "Kernel operators"  ''Proc. Kon. Nederl. Akad. Wetensch.'' , '''A 82'''  (1979)  pp. 39–53</TD></TR></table>