Compactness properties of weighted summation operators on trees

We investigate compactness properties of weighted summation operators V-alpha,V-sigma as mappings from l(1)(T) into l(q)(T) for some q is an element of (1, infinity). Those operators are defined by (V(alpha,sigma)x)(t) := alpha(t) Sigma(s >= t)sigma(s)x(s), t is an element of T, where T is a tree with partial order <=. 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,V-sigma), the (dyadic) entropy numbers of V-alpha,V-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.