On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example
✍ Scribed by J.F Toland; D Williams
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 397 KB
- Volume
- 160
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel between L 2 -spaces is deduced from an observation, Proposition 1.2, about the central role played by L 2 -spaces in the general theory of these operators.
Suppose (0, M, +) is a measure space and that K : 0_0 Ä [0, ) is an M_Mmeasurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded on any reasonable normed linear space X of M-measurable functions only if it is bounded on L 2 (0, M, +) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder ``Bounded Integral Operators on L 2 -Spaces,'' Springer-Verlag, BerlinÂNew York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an M-measurable (not necessarily square-integrable) function h>0 +-almosteverywhere on 0 with
implies that K is a bounded (self-adjoint) operator on L 2 (0, M, +) of norm at most 4. When (0, M, +) is _-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of K on L 2 (0, M, +) implies, for any 4>&K& 2 , the existence of a function h # L 2 (0, M, +) which is positive +-almost-everywhere and satisfies ( V ).
Such functions h satisfying ( V ), whether in L 2 (0, M, +) 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 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. A 69 (14) (1970Â1971), 199 204) on non-self-adjoint operators is derived in this way).
They may even act between different 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 different L 2 -spaces which arises naturally in Wiener Hopf theory and which has several puzzling features.