The Dunford-Pettis property on tensor products

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0\hat{\otimes}_\pi c_0)^{**}$ fails the DPP. Since $(c_0\hat{\otimes}_\pi c_0)^{*}$ does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$ are ${\mathscr L}_1$-spaces, then $E\hat{\otimes}_\epsilon F$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.Comment: 9 page

Surjective factorization of holomorphic mappings

We characterize the holomorphic mappings $f$ between complex Banach spaces that may be written in the form $f=T\circ g$, where $g$ is another holomorphic mapping and $T$ belongs to a closed surjective operator ideal.Comment: 8 page

The higher topological complexity of subcomplexes of products of spheres---and related polyhedral product spaces

We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.Comment: 30 pages. This second version of the paper extends the results of the first version to the case of polyhedral product spaces $Z(K,\{(S^{k_i},\star)\})$ where no restriction is assumed on the sphere dimensions $k_i