Compactness Properties of Weighted Summation Operators on Trees

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 $$(V_{\alpha ,\sigma }x)(t):=\alpha (t)\sum _{s⪰t}\sigma (s)x(s),t\in T,$$ where $T$ is a tree with induced partial order $t⪯s$ (or $s⪰t$) 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.