Compactness properties of weighted summation operators on trees-the critical case
The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill this gap. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed by the first-named author for a particular critical operator on the binary tree. Those problems appeared in a natural way during the study of compactness properties of certain Volterra integral operators in a critical case.