Compactness properties of weighted summation operators on trees
Tom 202 / 2011
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.
