85 research outputs found

### Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators

We study tensor norms that destroy unconditionality in the following sense:
for every Banach space $E$ with unconditional basis, the $n$-fold tensor
product of $E$ (with the corresponding tensor norm) does not have unconditional
basis. We establish an easy criterion to check weather a tensor norm destroys
unconditionality or not. Using this test we get that all injective and
projective tensor norms different from $\varepsilon$ and $\pi$ destroy
unconditionality, both in full and symmetric tensor products. We present
applications to polynomial ideals: we show that many usual polynomial ideals
never enjoy the Gordon-Lewis property. We also consider the unconditionality of
the monomial basic sequence. Analogous problems for multilinear and operator
ideals are addressed.Comment: 23 page

### Extending polynomials in maximal and minimal ideals

Given an homogeneous polynomial on a Banach space $E$ belonging to some
maximal or minimal polynomial ideal, we consider its iterated extension to an
ultrapower of $E$ and prove that this extension remains in the ideal and has
the same ideal norm. As a consequence, we show that the Aron-Berner extension
is a well defined isometry for any maximal or minimal ideal of homogeneous
polynomials. This allow us to obtain symmetric versions of some basic results
of the metric theory of tensor products.Comment: 13 page

### Bounded holomorphic functions attaining their norms in the bidual

Under certain hypotheses on the Banach space $X$, we prove that the set of
analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and
uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions
attain their norms, is dense in $\mathcal{A}_u(X)$. The result holds also for
functions with values in a dual space or in a Banach space with the so-called
property $(\beta)$. For this, we establish first a Lindenstrauss type theorem
for continuous polynomials. We also present some counterexamples for the
Bishop-Phelps theorem in the analytic and polynomial cases where our results
apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

### The symmetric Radon-Nikod\'ym property for tensor norms

We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for
finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type
theorem for s-tensor norms with this property. As a consequence, if $\beta$ is
a projective s-tensor norm with the sRN property, then for every Asplund space
$E$, the canonical map $\widetilde{\otimes}_{\beta}^{n,s} E' \to
\Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)'$ is a metric surjection. This
can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) =
\mathcal{Q}(E)$ for certain polynomial ideal \Q. We also relate the sRN
property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties
of different tensor products. Similar results for full tensor products are also
given. As an application, results concerning the ideal of $n$-homogeneous
extendible polynomials are obtained, as well as a new proof of the well known
isometric isomorphism between nuclear and integral polynomials on Asplund
spaces.Comment: 17 page

### Holomorphic Functions and polynomial ideals on Banach spaces

Given \u a multiplicative sequence of polynomial ideals, we consider the
associated algebra of holomorphic functions of bounded type, H_{b\u}(E). We
prove that, under very natural conditions verified by many usual classes of
polynomials, the spectrum M_{b\u}(E) of this algebra "behaves" like the
classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded
type holomorphic functions). More precisely, we prove that M_{b\u}(E) can be
endowed with a structure of Riemann domain over $E"$ and that the extension of
each f\in H_{b\u}(E) to the spectrum is an \u-holomorphic function of
bounded type in each connected component. We also prove a Banach-Stone type
theorem for these algebras.Comment: 19 page

### Non-linear Plank Problems and polynomial inequalities

We study lower bounds for the norm of the product of polynomials and their
applications to the so called \emph{plank problem.} We are particularly
interested in polynomials on finite dimensional Banach spaces, in which case
our results improve previous works when the number of polynomials is large.Comment: 19 page

### An integral formula for multiple summing norms of operators

We prove that the multiple summing norm of multilinear operators defined on
some $n$-dimensional real or complex vector spaces with the $p$-norm may be
written as an integral with respect to stables measures. As an application we
show inclusion and coincidence results for multiple summing mappings. We also
present some contraction properties and compute or estimate the limit orders of
this class of operators.Comment: 19 page

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