Compactness properties of weighted summation operators on trees

Mikhail Lifshits, and Werner Linde. "Compactness properties of weighted summation operators on trees." Studia Mathematica 202.1 (2011): 17-47. <http://eudml.org/doc/285501>.

@article{MikhailLifshits2011,
abstract = {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.},
author = {Mikhail Lifshits, Werner Linde},
journal = {Studia Mathematica},
keywords = {metrics on trees; operators on trees; weighted summation operators; covering numbers; entropy numbers},
language = {eng},
number = {1},
pages = {17-47},
title = {Compactness properties of weighted summation operators on trees},
url = {http://eudml.org/doc/285501},
volume = {202},
year = {2011},
}

TY - JOUR
AU - Mikhail Lifshits
AU - Werner Linde
TI - Compactness properties of weighted summation operators on trees
JO - Studia Mathematica
PY - 2011
VL - 202
IS - 1
SP - 17
EP - 47
AB - 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.
LA - eng
KW - metrics on trees; operators on trees; weighted summation operators; covering numbers; entropy numbers
UR - http://eudml.org/doc/285501
ER -