On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

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,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 onL2(Ω,M,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 M-measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithKh(x)=∫ΩK(x,y)h(y)μ(dy)⩽Λh(x)(x∈Ω) (*)implies thatKis a bounded (self-adjoint) operator onL2(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 onL2(Ω,M,μ) implies, for anyΛ>‖K‖2, the existence of a functionh∈L2(Ω,M,μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL2(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 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.