We study Kalton's theorem on the unconditional convergence of series of compact operators and we use some matrix techniques to obtain sufficient conditions, weaker than the previous one, on the convergence and unconditional convergence of series of compact operators.

In this note we investigate the relationship between the convergence of the sequence $\left\{S_n\right\}$ of sums of independent random elements of the form $S_n={\sum }_{i=1}^n{\epsilon }_i{x}_i$ (where ${\epsilon }_i$ takes the values $\pm 1$ with the same probability and $x_i$ belongs to a real Banach space $X$ for each $i\in \mathbb{N}$) and the existence of certain weakly unconditionally Cauchy subseries of ${\sum }_{n=1}^{\infty }{x}_n$.

On every subspace of $l_{\infty }\left(\mathbb{N}\right)$ which contains an uncountable $\omega$-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_{\infty }\left(\mathbb{N}\right)$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.

Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X₁,...,Xₙ are independent copies of X, then $1/C_p{\left|\right|x\left|\right|}_M\le \left|\right|\left(x_i{X}_i\right){ⁿ}_{i=1}{\left|\right|}_p\le C_p{\left|\right|x\left|\right|}_M$, where $C_p$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the ${\ell }_p$-norm by an arbitrary N-norm. This...

We give sufficient conditions on Banach spaces E and F so that their projective tensor product $E{\otimes }_{\pi }F$ and the duals of their projective and injective tensor products do not have the Dunford-Pettis property. We prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:E* → F** is completely continuous, then $\left(E{\otimes }_{ϵ}F\right)*$ does not have the DPP. We also prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T: F** → E* is completely...

In [K-S 1] it was shown that $Av{e}_{\pi }({\sum }_{i=1}^n|x_i{a}_{\pi \left(i\right)}{{|}^2)}^{1/2}$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1,...,a_n$ so that the above expression is equivalent to a given Orlicz norm.

We establish some sufficient conditions under which the subspaces of Dunford-Pettis operators, of M-weakly compact operators, of L-weakly compact operators, of weakly compact operators, of semi-compact operators and of compact operators coincide and we give some consequences.