Compactness properties of weighted summation operators on trees

• Autorzy: Mikhail Lifshits , Werner Linde
• Języki publikacji: EN

Abstrakty

EN

We investigate compactness properties of weighted summation operators $V_{α,σ}$ as mappings from ℓ₁(T) into $ℓ_{q}(T)$ for some q ∈ (1,∞). Those operators are defined by
$(V_{α,σ}x)(t) : = α(t) ∑_{s⪰t} σ(s)x(s)$, 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.

Kategorie tematyczne

• 05C05: Trees
• 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s -numbers, Kolmogorov numbers, entropy numbers, etc. of operators
• 06A06: Partial order, general

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 202
• Numer: 1
• Strony: 17-47

Daty

wydano

2011

Twórcy

autor

Mikhail Lifshits

• Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl. 2, 198504 Stary Peterhof, Russia

autor

Werner Linde

• Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany

Identyfikatory

DOI

10.4064/sm202-1-2