### $\u2102$-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation.

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F\u0303{x}^{(2n+1)}$ where $F\u0303:{X}^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

We give an example of a fourth degree polynomial which does not satisfy Rolle’s Theorem in the unit ball of ${l}_{2}$.

We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $ma{x}_{\omega \in}\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$. We study the stability of the APDP by c₀-, ${\ell}_{\infty}$- and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely ${L}_{\infty}(\mu ,X)$ and C(K,X), where X has the APDP.

Given an operator ideal ℐ, a Banach space E has the ℐ-approximation property if the identity operator on E can be uniformly approximated on compact subsets of E by operators belonging to ℐ. In this paper the ℐ-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of linear operators/homogeneous polynomials, spaces of holomorphic functions and their preduals.

We present simple proofs that spaces of homogeneous polynomials on ${L}_{p}[0,1]$ and ${\ell}_{p}$ provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

Using axiomatic joint spectra we obtain a functional calculus which extends our previous Gelfand-Waelbroeck type results to include a Banach-valued Taylor-Waelbroeck spectrum.

The purpose of this note is to announce, without proofs, some results concerning vector valued multilinear operators on a product of C(K) spaces.

We give a unified treatment of procedures for complexifying real Banach spaces. These include several approaches used in the past. We obtain best possible results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. These estimates provide generalizations and show sharpness of previously obtained inequalities.

In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

We lift to homogeneous polynomials and multilinear mappings a linear result due to Lindenstrauss and Pełczyński for absolutely summing operators. We explore the notion of cotype to obtain stronger results and provide various examples of situations in which the space of absolutely summing homogeneous polynomials is different from the whole space of homogeneous polynomials. Among other consequences, these results enable us to obtain answers to some open questions about absolutely summing homogeneous...

Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {\u0302}_{n,\pi}E\right)$ (resp. $\Delta \left(\otimes {\u0302}_{n,s,\pi}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {\u0302}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {\u0302}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to ${E}_{\left[n\right]}$, the completion of the n-concavification of...

We prove that extendible 2-homogeneous polynomials on spaces with cotype 2 are integral. This allows us to find examples of approximable non-extendible polynomials on ${\ell}_{p}$ (1 ≤ p < ∞ ) of any degree. We also exhibit non-nuclear extendible polynomials for 4 < p < ∞. We study the extendibility of analytic functions on Banach spaces and show the existence of functions of infinite radius of convergence whose coefficients are finite type polynomials but which fail to be extendible.