Compactness Properties of Weighted Summation Operators on Trees

We investigate compactness properties of weighted summation operators $V_{α,σ}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $(V_{α,σ} x)(t) :=α(t)\sum_{s\succeq t}σ(s) x(s)\,,\quad t\in T\;,$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $α$ and $σ$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{α,σ})$, the (dyadic) entropy numbers of $V_{α,σ}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $α(t)σ(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees.

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