On the Schur test for L-2-boundedness of positive integral operators with a Wiener-Hopf example

The Schur sufficiency condition for boundedness of any integral operator wi th non-negative kernel between L-2-spaces is deduced from an observation, P roposition 1.2, about the central role played by L-2-spaces in the general theory of these operators. Suppose (Omega, M, mu) is a measure space and that K: Omega x Omega --> [0, infinity) is an M x M-measurable kernel. The special case of Proposition 1 .2 for symmetrical kernels says that such a linear integral operator is bou nded on any reasonable normed linear space X of M-measurable functions only if it is bounded on L-2(Omega, M, mu) where its norm is no larger. The gen eral form of Schur's condition (Halmos and Sunder "Bounded Integral Operato rs on L-2-Spaces," Springer-Verlag, Berlin/New York, 1978) is a simple coro llary which, in the symmetrical case, says that the existence of an M-measu rable (not necessarily square-integrable) function h>0 mu-almost-everywhere on Omega with Kh(x) = integral(Omega) K(x, y) h(y) mu(dy) less than or equal to Lambda h( x) (x is an element of Omega) (*) implies that K is a bounded (self-adjoint) operator on L-2(Omega, M, mu) of norm at most Lambda. When (Omega, M ,mu) is sigma-finite, we show that Sch ur's condition is sharp: in the symmetrical case the boundedness of K on L- 2(Omega, M, mu) implies, for any Lambda > parallel to K parallel to(2), the existence of a function h is an element of L-2(Omega, M, mu), which is pos itive mu-almost-everywhere and satisfies(*). Such functions h satisfying (*), whether in L-2(Omega, M, mu) or not, will be called Schur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of inte gral operators with non-negative kernels. In the general theory the operato rs are not required to be symmetrical (a theorem of Chisholm and Everitt (P roc. Rov. Sue. Edinburgh Sect. A 69 (14) (1970/1971), 199 204) on non-self- adjoint operators is derived in this way). They may even act between differ ent L-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 differe nt L-2-spaces which arises naturally in Wiener-Hopf theory and which has se veral puzzling features. (C) 1998 Academic Press.