On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example |
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Authors: | JF Toland D Williams |
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Institution: | Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdomf1 |
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Abstract: | The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL2-spaces is deduced from an observation, Proposition 1.2, about the central role played byL2-spaces in the general theory of these operators. Suppose (Ω, , μ) is a measure space and thatK: Ω×Ω→0, ∞) is an × -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 -measurable functions only if it is bounded onL2(Ω, , μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder “Bounded Integral Operators onL2-Spaces,” Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an -measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwith implies thatKis a bounded (self-adjoint) operator onL2(Ω, , μ) of norm at mostΛ. When (Ω, , μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of onL2(Ω, , μ) implies, for anyΛ>‖ ‖2, the existence of a functionh∈L2(Ω, , μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL2(Ω, , 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 differentL2-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL2-spaces which arises naturally in Wiener–Hopf theory and which has several puzzling features. |
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