## Abstract

The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL_{2}-spaces is deduced from an observation, Proposition 1.2, about the central role played byL_{2}-spaces in the general theory of these operators. Suppose (Ω,M,μ) is a measure space and thatK:Ω×Ω→[0,∞) is an M×M-measurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded onanyreasonable normed linear spaceXof M-measurable functions only if it is bounded onL_{2}(Ω,M,μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder "Bounded Integral Operators onL_{2}-Spaces," Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an M-measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithKh(x)=∫ _{Ω}K(x,y)h(y)μ(dy)≤Λh(x)(x∈Ω) (*)implies thatKis a bounded (self-adjoint) operator onL_{2}(Ω,M,μ) of norm at mostΛ. When (Ω,M,μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of K onL_{2}(Ω,M,μ) implies, for anyΛ>K_{2}, the existence of a functionh∈L_{2}(Ω,M,μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL_{2}(Ω,M,μ) or not, will be calledSchur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of integral operators with non-negative kernels. In the general theory the operators are not required to be symmetrical (a theorem of Chisholm and Everitt (Proc. Roy. Soc. Edinburgh Sect. A69(14) (1970/1971), 199-204) on non-self-adjoint operators is derived in this way). They may even act between differentL_{2}-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL_{2}-spaces which arises naturally in Wiener-Hopf theory and which has several puzzling features.

Original language | English |
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Pages (from-to) | 543-560 |

Number of pages | 18 |

Journal | Journal of Functional Analysis |

Volume | 160 |

Issue number | 2 |

DOIs | |

Publication status | Published - 20 Dec 1998 |

## ASJC Scopus subject areas

- Analysis

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