We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without $c_0$ copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the $c_0$-theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in $c_0$.

Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_m$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_m$. Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also consider the...

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:L_q\left(\mu \right)\to L_p\left(\nu \right)$, each n ∈ ℕ and functions $f_1,...,f_n\in L_q\left(\mu \right)$, $\left(ʃ\right({\sum }_{k=1}^n|T{f}_k{|}^r{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(ʃ\right({\sum }_{k=1}^n|f_k{|}^r{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...