Titlebook: Bounded Integral Operators on L 2 Spaces; Paul Richard Halmos,Viakalathur Shankar Sunder Book 1978 Springer-Verlag Berlin Heidelberg 1978

只看該作者 · 發(fā)表于 2025-3-24 03:24:21

Measure Spaces,A matrix is a function. A complex . × . (rectangular) matrix, for example, is a function . from the Cartesian product {1,...,.} × {1,...,.} to the set ? of complex numbers; its value at the ordered pair <., .> is usually denoted by ... In this book it will always be denoted by the typographically and conceptually more convenient symbol .(., .).

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Domains,The way a matrix acts is defined by the familiar formula . The generalization to arbitrary kernels is formally obvious: . Finite sums such as the ones in (1) can always be formed; integrals such as theones indicated in (2) may fail to exist and, even when they exist, may fail todefine well-behaved functions.

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Examples,The easiest examples of bounded kernels are the square-integrable ones introduced in Lemma 4.1; they induce Hilbert-Schmidt operators. The examples that follow are different; they are, for one thing, not compact.

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只看該作者 · 發(fā)表于 2025-3-24 20:52:06

Carleman Kernels,There is a sense in which the most natural integral operators on .. are the ones induced by Carleman kernels (the semi-square-integrable kernels ., for which .(.,·)∈ ..(.) for almost every .).

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