Compactness properties of weighted summation operators on trees

Abstract

We investigate compactness properties of weighted summation operators Vα, σ as mappings from l1(T) into l q(T) for some q ∈ (1, ∞). Those operators are defined by (Vα,σx)(t) := α(t) Σs?t σ(s)x(s), t ∈ T, where T is a tree with partial order . 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 en(V α,σ), the (dyadic) entropy numbers of V α,σ. The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t) decreasing either polynomially or exponentially. We also give some probabilistic applications to Gaussian summation schemes on trees. © Instytut Matematyczny PAN, 2011.