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Hille-Tamarkin operator

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Let be an integral operator from into , i.e., there exists a -measurable function on such that a.e. on . Then is called a Hille–Tamarkin operator if

where . By taking one obtains the class of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator). Replacing , respectively , by arbitrary Banach function norms , respectively , one obtains the class of Hille–Tamarkin operators between Banach function spaces (sometimes called integral operators of finite double norm, see [a1]). Under some mild hypotheses on the norms, one can show that Hille–Tamarkin operators have rather strong compactness properties (see [a2]).

References

[a1] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a2] A.R. Schep, "Compactness properties of Carleman and Hille–Tamarkin operators" Canad. J. Math. , 37 (1985) pp. 921–933
How to Cite This Entry:
Hille-Tamarkin operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hille-Tamarkin_operator&oldid=19283
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article