arXiv subjects

Joaquín M. Gutiérrez

Publications and source records attributed to Joaquín M. Gutiérrez.

5 recordsLinked to original sources

arXiv2000

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 $(c_0\hat{\otimes}_πc_0)^{**}$ fails the DPP. Since $(c_0\hat{\otimes}_πc_0)^{*}$ 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 $E\hat{\otimes}_εF$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.

math.FA↗

arXiv2000

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.

math.FA↗

arXiv2000

The polynomial property (V)

Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on bounded sets. It is shown that if E has an unconditional finite dimensional expansion of the identity, the following assertions are equivalent: (a) ${\mathcal P}(^k!E,F)={\mathcal P}_{wb}(^k!E,F)$; (b) ${\mathcal P}_{wb}(^k!E,F)$ contains no copy of $c_0$; (c) ${\mathcal P}(^k!E,F)$ contains no copy of $\ell_\infty$; (d) ${\mathcal P}_{wb}(^k!E,F)$ is complemented in ${\mathcal P}(^k!E,F)$. This result was obtained by Kalton for linear operators. As an application, we show that if E has Pełczyński's property (V) and satisfies ${\mathcal P}(^k!E) ={\mathcal P}_{wb}(^k!E)$ then, for all F, every unconditionally converging $P\in{\mathcal P}(^k!E,F)$ is weakly compact. If E has an unconditional finite dimensional expansion of the identity, then the converse is also true.

math.FA↗

arXiv2000

Orlicz-Pettis polynomials on Banach spaces

We introduce the class of Orlicz-Pettis polynomials between Banach spaces, defined by their action on weakly unconditionally Cauchy series. We give a number of equivalent definitions, examples and counterexamples which highlight the differences between these polynomials and the corresponding linear operators.

math.FA↗

arXiv2000

Polynomials on Schreier's space

We introduce a weakened version of the Dunford-Pettis property, and give examples of Banach spaces with this property. In particular, we show that every closed subspace of Schreier's space $S$ enjoys it. As an application, we characterize the weak polynomial convergence of sequences, show that every closed subspace of $S$ has the polynomial Dunford-Pettis property of Biström et al. and give other polynomial properties of $S$.

math.FA↗