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Integral operators and absolute convergence systems. (Russian, English) Zbl 1016.45008

Sib. Mat. Zh. 43, No. 3, 573-578 (2002); translation in Sib. Math. J. 43, No. 3, 458-462 (2002).
The author uses special integral operators (Carleman operators) to establish several special properties of absolute convergence systems for \(l_2\).
The following notion of an absolute convergence system is used: a sequence \(\{g_n\}\subset M\) (\(M = M(X,\mu)\) denotes the space of all \(\mu\)-measurable \(\mu\)-a.e. finite functions on \(X\), where \((X,\mu)\) is a measure space with \(\sigma\)-finite positive nonatomic measure) is an absolute convergence system for \(l_2\) if for every sequence \(\{a_n\}\in l_2\) the series \(\sum_{n=1}^{\infty}|a_ng_n(s)|\) converges \(\mu\)-a.e.; furthermore, the convergence set of this series depends on \(\{a_n\}\).
The author gives a criterion which guarantees that sequences under the study are absolute convergence systems for \(l_2\) and exposes a result on generating integral operators by absolute convergence systems.

MSC:

45P05 Integral operators
47G10 Integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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