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

Mikhail Lifshits¹ ; Werner Linde²

¹ Department of Mathematics and Mechanics St. Petersburg State University Bibliotechnaya pl. 2 198504 Stary Peterhof, Russia
² Institut für Stochastik Friedrich-Schiller-Universität Jena Ernst-Abbe-Platz 2 07743 Jena, Germany

Studia Mathematica, Tome 206 (2011) no. 1, pp. 75-96

Résumé

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.

Détail

DOI : 10.4064/sm206-1-6

Keywords: paper provide upper bounds entropy numbers summation operators trees critical recent paper studia math nbsp elaborated framework weighted summation operators general trees where related entropy operator those underlying tree equipped appropriate metric however results incomplete critical entropy behavior because requires much involved techniques present article fill gap end develop method working context general trees general weighted summation operators which recently proposed first named author particular critical operator binary tree those problems appeared natural during study compactness properties certain volterra integral operators critical