Compactness properties of weighted summation operators on trees - the critical case

TL;DR

This paper completes the analysis of entropy numbers for weighted summation operators on trees in a critical case, using advanced techniques to fill gaps from previous incomplete results, with applications to Volterra integral operators.

Contribution

It introduces a new method to analyze the critical case of entropy behavior for summation operators on general trees, extending previous work and filling existing gaps.

Findings

Provided upper bounds for entropy numbers in the critical case

Developed a general method applicable to various trees and operators

Connected results to compactness properties of Volterra integral operators

Abstract

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 [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with 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 the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case.