# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Compactness properties of weighted summation operators on trees

### Tom 202 / 2011

Studia Mathematica 202 (2011), 17-47 MSC: Primary 47B06; Secondary 06A06, 05C05. DOI: 10.4064/sm202-1-2

#### Streszczenie

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.

#### Autorzy

• Mikhail LifshitsDepartment of Mathematics and Mechanics
St. Petersburg State University
Bibliotechnaya pl. 2
198504 Stary Peterhof, Russia
e-mail
• Werner LindeInstitut für Stochastik
Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2
07743 Jena, Germany
e-mail

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