Extension and Lifting of Operators and Polynomials

Castillo, Jesús M. F.; García, Rubén J.; Suárez, Jesús

doi:10.1007/s00009-011-0150-8

Cited by 4 publications

(3 citation statements)

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“…Results on lifting of polynomials (and holomorphic functions) can be traced back to [2], [33] and much research on this topic has been done since then, see, e.g. [13], [14], [20]. We prove some new results on lifting of polynomials, including quantitative refinements of known results, we shall use later.…”

Section: Lifting Of Polynomialsmentioning

confidence: 80%

The surjective hull of a polynomial ideal

2016

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The aim of this paper is the study of surjective ideals of homogeneous polynomials between Banach spaces. To do so we define the surjective hull of a polynomial ideal and prove the main properties of this hull procedure. For a more comprehensive theory, new lifting properties of homogeneous polynomials are proved and applied to the description of the surjective hulls of the ideals of scriptI‐bounded polynomials and of composition polynomials ideals. Several applications are provided.

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Section: Lifting Of Polynomialsmentioning

confidence: 80%

The surjective hull of a polynomial ideal

2016

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“…This immediately implies that bilinear forms on L 1 subspaces of L 1 spaces extend to the bigger space (as well as bilinear form on subspaces of L ∞ spaces inducing L ∞ -quotients) ( [14,29]). And all bilinear forms on Y extend to…”

Section: Definition 1 ([29]mentioning

confidence: 99%

Local complementation and the extension of bilinear mappings

Castillo¹,

García²,

Defant³

et al. 2011

Math. Proc. Camb. Phil. Soc.

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We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if $X$ is not a Hilbert space then one may find a subspace of $X$ for which there is no Aron-Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given $X$ forces such $X$ to contain no uniform copies of $\ell_p^n$ for $p\in[1,2)$. In particular, $X$ must have type $2-\epsilon$ for every $\epsilon>0$. Also, we show that the bilinear version of the Lindenstrauss-Pe{\l}czy\'nski and Johnson-Zippin theorems fail. We will then consider the notion of locally $\alpha$-complemented subspace for a reasonable tensor norm $\alpha$, and study the connections between $\alpha$-local complementation and the extendability of $\alpha^*$ -integral operators.Comment: to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

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“…Let us sum up known facts• The derived functor of an Exact functor should be 0.• Ext Ban is the derived functor of L : Ban −→ Vect.• The derived functor of ⊗ is called Tor.• The derivation of operator ideal functors A opens the topic of relative homology[19] • Derivation continues, and the derived functor of Ext Ban is called Ext 2Ban . In general, the derived functor of Ext n Ban is denoted Ext n+1 Ban .…”

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confidence: 99%