Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Garcı́a, Domingo; Grecu, Bogdan C.; Maestre, Manuel

doi:10.1007/s00605-008-0017-7

Cited by 3 publications

(4 citation statements)

References 19 publications

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“…Namely, if X is a Banach spaces whose dual is separable and has the approximation property, we see in Theorem 2.2 that any element in the tensor product (˜ N,s π s X)˜ π Y is associated to a regular Borel measure on (B X , w * ) × (B Y , w * ), for any Banach space Y . This integral formula somehow extends those given in [19,20]. In Theorem 2.3, we apply our integral representation to prove a Lindenstrauss theorem for homogeneous polynomials from Banach spaces X satisfying the hypotheses above, into any dual space (and, therefore, for scalar-valued homogeneous polynomials on X).…”

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confidence: 88%

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On the polynomial Lindenstrauss theorem

Carando

Lassalle

Mazzitelli

2012

Journal of Functional Analysis

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Under certain hypotheses on the Banach space $X$, we show that the set of $N$-homogeneous polynomials from $X$ to any dual space, whose Aron-Berner extensions are norm attaining, is dense in the space of all continuous $N$-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop-Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollob\'as, of these results

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confidence: 88%

“…Now we prove the integral representation for the elements in the tensor product (˜ N,s π s X)˜ π Y , which should be compared with [20, Theorem 1] and [19,Remark 3.6]. As usual, we consider B X and B Y endowed with their weak-star topologies, which make them compact sets.…”

Section: Integral Representation Of Tensors and The Polynomial Lindenmentioning

confidence: 98%

On the polynomial Lindenstrauss theorem

Carando

Lassalle

Mazzitelli

2012

Journal of Functional Analysis

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“…The investigation of polynomials and multilinear operators acting on Banach spaces is a fruitful topic of investigation that dates back to the 30 ′ s (see, for instance [10,19,20] and, for recent papers, [7,8,11,15,17] among many others).…”

Section: Introductionmentioning

confidence: 99%

Polynomial and multilinear Hardy-Littlewood inequalities: analytical and numerical approaches

Campos¹,

Cavalcante²,

Fávaro³

et al. 2018

Mathematical Inequalities &Amp; Applications

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Abstract. We investigate the constants of the polynomial and multilinear Hardy-Littlewood inequalities. Among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araújo et al. In a final section, as an independent appendix, we present some computer-aided estimates for the lower bounds of the multilinear Hardy-Littlewood inequalities.Mathematics subject classification (2010): 46G25, 47L22, 47H60.

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“…translation of the linear one, neither when it comes to algebraic nor analytical or geometrical properties [1,2,9,13].…”

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confidence: 99%

The unit ball of the complex $${{\mathcal P}(^3H)}$$

2008

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Let H be a two-dimensional complex Hilbert space and P( 3 H ) the space of 3-homogeneous polynomials on H . We give a characterization of the extreme points of its unit ball, B P( 3 H ) , from which we deduce that the unit sphere of P( 3 H ) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B P( 3 H ) remains extreme as considered as an element of B L( 3 H ) . Finally we make a few remarks about the geometry of the unit ball of the predual of P( 3 H ) and give a characterization of its smooth points. Notation, terminology and preliminary resultsGiven n (real or complex) Banach spaces X 1 , . . . , X n we denote by X 1 ⊗· · ·⊗ X n their tensor product and by π the projective norm. If X 1⊗π · · ·⊗ π X n is the completion of X 1 ⊗· · · ⊗ X n under the projective norm, then we have (X 1⊗π · · ·⊗ π X n ) * = L( n X 1 , . . . , X n ), the space of continuous n-linear forms on X 1 × · · · × X n endowed with the supremum norm. However, as remarked in [6], most of the times the multilinear theory is far from being just a simple

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Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Cited by 3 publications

References 19 publications

On the polynomial Lindenstrauss theorem

On the polynomial Lindenstrauss theorem

Polynomial and multilinear Hardy-Littlewood inequalities: analytical and numerical approaches

The unit ball of the complex $${{\mathcal P}(^3H)}$$

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