When every polynomial is unconditionally converging

Abstract

Letting $E$, $F$ be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator $E\rightarrow F$ is unconditionally converging, then every polynomial from $E$ to $F$ is unconditionally converging (definition as in the linear case). (2) If $E$ has the Dunford-Pettis property and every operator $E\rightarrow F$ is weakly compact, then every $k$-linear mapping from $E^k$ into $F$ takes weak Cauchy sequences into norm convergent sequences. In particular, every polynomial from $\ell_\infty$ into a space containing no copy of $\ell_\infty$ is completely continuous. This solves a problem raised by the authors in a previous paper, where they showed that there exist nonweakly compact polynomials from $\ell_\infty$ into any nonreflexive space.