This is an announcement for the paper "Compactness properties of weighted summation operators on trees" by Mikhail Lifshits and Werner Linde.
Abstract: We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$ (V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s),,\quad t\in T;, $$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g.~moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees.
Archive classification: math.FA
Mathematics Subject Classification: Primary: 47B06, Secondary: 06A06, 05C05
Submitted from: lifts@mail.rcom.ru
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1006.3867
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