| On the Schur Test for L 2 -Boundedness of Positive Integral Operators and a Wiener-Hopf Example (2007) | |||||||||||||||
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| 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(\Omega ; M;) is a measure space and that K :\Omega \Theta\Omega ! [0; 1) is an M \Theta M-measurable 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 if, and only if, it is bounded on L 2(\Omega ; M;) where its norm is no larger. The general form of Schur's condition [2] is a simple corollary which, in the symmetrical case, says that the existence of an M-measurable (not necessarily square-integrable) function h ? 0 -almost-everywhere on \Omega with Kh(x) = Z \Omega K(x; y)h(y)(dy) h(x) (x 2 \Omega\Gamma () implies that K is a bounded (self-adjoint) operator on L 2(\Omega ; M;) of n... | |||||||||||||||
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