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....

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 $\left(V_{\alpha ,\sigma }x\right)\left(t\right):=\alpha \left(t\right){\sum }_{s⪰t}\sigma \left(s\right)x\left(s\right)$, 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 $eₙ\left(V_{\alpha ,\sigma }\right)$, 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)...

We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${\left({\sum }_k{(\left(\parallel T{x}_k{\parallel }_F\right)/\left(\surd log(k+1)\right))}^q\right)}^{1/q}\le c\parallel {\sum }_k{\varepsilon }_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $({\sum }_k{\left(\parallel T{x}_k{\parallel }_F\surd \left({\left(log(k+1)\right)}^q\right)\right)}^{1/q}\le c\parallel {\sum }_k{g}_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

In this paper we first introduce the concept of compatible mappings of type (B) and compare these mappings with compatible mappings and compatible mappings of type (A) in Saks spaces. In the sequel, we derive some relations between these mappings. Secondly, we prove a coincidence point theorem and common fixed point theorem for compatible mappings of type (B) in Saks spaces.

The object of this paper is to establish a unique common fixed point theorem for six self-mappings satisfying a generalized contractive condition through compatibility of type $\left(\beta \right)$ and weak compatibility in a fuzzy metric space. It significantly generalizes the result of Singh and Jain [The Journal of Fuzzy Mathematics $\left(2006\right)]$ and Sharma [Fuzzy Sets and Systems $\left(2002\right)]$. An example has been constructed in support of our main result. All the results presented in this paper are new.