Integral Operators and Absolute Convergence Systems
Copyright policy )Integral Operators and Absolute Convergence Systems
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.
- Sobolev Institute of Mathematics Russian Federation
Integral operators, absolute convergence system, integral operator, Carleman operator, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Integral operators, absolute convergence system, integral operator, Carleman operator, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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