### Moments of Measures on Banach Spaces.

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Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T={t}_{1},{t}_{2},...$, $\left|\right|{t}_{j}\left|\right|\le {a}_{j}$, by functions of the ${a}_{j}$’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ${\left({a}_{j}\right)}_{j=1}^{\infty}$.

We investigate compactness properties of weighted summation operators ${V}_{\alpha ,\sigma}$ as mappings from ℓ₁(T) into ${\ell}_{q}\left(T\right)$ for some q ∈ (1,∞). Those operators are defined by $\left({V}_{\alpha ,\sigma}x\right)\left(t\right):=\alpha \left(t\right){\sum}_{s\u2ab0t}\sigma \left(s\right)x\left(s\right)$, 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 $e\u2099\left({V}_{\alpha ,\sigma}\right)$, the (dyadic) entropy numbers of ${V}_{\alpha ,\sigma}$. The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t)...

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....

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