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 $E\to F$ 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 $E\to F$ 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.

References in corpus (1)

• Unconditionally converging polynomials on Banach spaces