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Compactness properties of weighted summation operators on trees
Keywords: entropy numbers; summation operators; trees
Thu, 10:45--11:10
  • Lifshits, Mikhail (Dept Math.Mech., St.Petersburg State University; HIM (semester long term participant), Russia)
  • Linde, Werner (Friedrich-Schiller-Universität Jena)

We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$ (V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s \succeq t}\sigma(s) x(s)\,,\quad t\in T\;, $$ where $T$ is a tree and $\alpha$ and $\sigma$ are given weights on $T$. Such operators are natural discrete analogues of Volterra operators.

We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two–sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees.

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