Compactness properties of weighted summation operators on trees

Mikhail Lifshits ¹   ; Werner Linde ²

¹ Department of Mathematics and Mechanics St. Petersburg State University Bibliotechnaya pl. 2 198504 Stary Peterhof, Russia
² Institut für Stochastik Friedrich-Schiller-Universität Jena Ernst-Abbe-Platz 2 07743 Jena, Germany

Studia Mathematica, Tome 202 (2011) no. 1, pp. 17-47

Résumé

We investigate compactness properties of weighted summation operators $V_{\alpha ,\sigma }$ as mappings from $\ell _1(T)$ into $\ell _q(T)$ for some $q\in (1,\infty )$. Those operators are defined by $$ (V_{\alpha ,\sigma } x)(t) :=\alpha (t) \sum _{s\succeq t}\sigma (s) x(s),\hskip 1em t\in T, $$ where $T$ is a tree with partial order $\preceq $. 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 to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with $\alpha (t)\sigma (t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications to Gaussian summation schemes on trees.

Détail

DOI : 10.4064/sm202-1-2

Keywords: investigate compactness properties weighted summation operators alpha sigma mappings ell ell infty those operators defined alpha sigma alpha sum succeq sigma hskip where tree partial order preceq here alpha sigma given weights introduce metric compactness properties imply two sided estimates alpha sigma dyadic entropy numbers alpha sigma results applied concrete trees moderately increasing biased binary trees weights alpha sigma decreasing either polynomially exponentially probabilistic applications gaussian summation schemes trees