Unconditionality in Tensor Products and Ideals of Polynomials, Multilinear Forms and Operators

Let, ( ) be the best constant that fulfills the following inequality: for every -homogeneous polynomial ( ) = ∑ | |= in complex variables,For every degree , and a wide range of values of , ∈ [1, ∞] (including any in the case ∈ [1, 2], and any and for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as (the number of variables) tends to infinity.Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., -homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of , . As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality. K E Y W O R D SHardy-Littlewood inequalities, multivariable von Neumann's inequality, unconditionality in spaces of polynomials, unimodular polynomials M S C ( 2 0 1 0 ) 11C08, 15A60, 15A69, 46G25, 47A30, 47H60 Problem 1.3. Let , ( ) and , ( ) be the smallest constants that fulfill the following inequalities: for every -homogeneous polynomial in complex variables,

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“…The following result shows that, in order to study the asymptotic behavior of the mixed unconditional constants of

P {false(}^{m} {double-struckC}^{n})

, it is enough to understand what happens with the monomial basis

{false(z^{α} false)}_{α \in normalΛ false(m, n false)}

. These can be seen as a sort extension of a result of Pisier and Schütt (see also ). Theorem We have the following relation:

χ_{p, q} false(scriptP {false(}^{m} {double-struckC}^{n}) false) \leq χ_{p, q} ({(z^{α})}_{α \in Λ (m, n)}) \leq 2^{m} χ_{p, q} false(scriptP {false(}^{m} {double-struckC}^{n}) false) .

…”

Section: Mixed Unconditional Basis Constant For Homogeneous Polynomiamentioning

confidence: 73%

The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Galicer

Mansilla

Muro

2019

Mathematische Nachrichten

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“…], we have that P m Q(ℓ 2 ) is uniformly bounded. Since T does not contain (ℓ m 2 ) m nor (ℓ m ∞ ) m uniformly complemented (see [14, pages 33 and 66]), we can conclude that Q(T ) cannot be separable by [13,Proposition 4.9]. As a consequence, Q(T ) cannot coincide with Q min (T ).…”

Section: The Symmetric Radon-nikodým Propertymentioning

confidence: 97%

The symmetric Radon–Nikodým property for tensor norms

Carando¹,

Galicer²

2011

Journal of Mathematical Analysis and Applications

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We introduce the symmetric Radon-Nikodým property (sRN property) for finitely generated s-tensor norms β of order n and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β is a projective s-tensor norm with the sRN property, then for every Asplund space E, the canonical mappingsurjection. This can be rephrased as the isometric isomorphism Q min (E) = Q(E) for some polynomial ideal Q. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikodým properties of different tensor products. 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. An analogous study is carried out for full tensor products.A result of Boyd and Ryan [5] and also of Carando and Dimant [9] implies that, for an Asplund space E, the space P n I (E) of integral polynomials is isometric to the space P n N (E) of nuclear polynomials (the isomorphism between these spaces was previously obtained by Alencar in [1,2]). In other words, if E is Asplund, the space of integral polynomials on E coincides isometrically with its minimal hull (P n I ) min (E) = P n N (E). This fact was used, for example, in [5,6,17] to study geometric properties of spaces of polynomials and tensor products (e.g., extreme and exposed points of their unit balls), and in [7,8] to characterize isometries between spaces of polynomials and centralizers of symmetric tensor products. When the above mentioned isometry is stated as the isometric coincidence between a maximal ideal and its minimal hull, it resembles the Lewis theorem for operator ideals and (2-fold) tensor norms (see [24] and [15, 33.3]). The Radon-Nikodým property for tensor norms is a key ingredient for Lewis theorem.The aim of this article is to find conditions under which the equality Q(E) = Q min (E) holds isometrically for a maximal polynomial ideal Q. In terms of symmetric tensor products, we want conditions on an s-tensor norms ensuring the isometry n,s β E = ( n,s β E) . To this end, we introduce the symmetric Radon-Nikodým property for s-tensor norms and show our main result, a Lewis-type theorem (Theorem 2.2): if an s-tensor norm has the symmetric Radon-Nikodým property (sRN property), then the canonical mapping n,s \β/ E → ( n,s /β \ E) is a metric surjection for every Asplund space E (see the notation below). As a consequence, if Q is the maximal ideal (of n-homogeneous polynomials) associated to a projective s-tensor norm β with the sRN property, then Q min (E) = Q(E) isometrically.

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“…. , i n ) ∈ N n , i 1 · · · i n }, respectively, is not necessary an unconditional basis of⊗ n,ǫ X and⊗ n,s,ǫ X (see, e.g., [14,18,20,8,17,6]). Next we will use the following Rademacher averaging formula (see, e.g., [19,Lemma 2.22]) to show that the tensor diagonal {e i ⊗ · · · ⊗ e i : i ∈ N} is an unconditional basis of both ∆(⊗ n,ǫ X) and ∆(⊗ n,s,ǫ X), and their diagonal projections are contractive.…”

Section: Diagonals Of Injective Tensor Productsmentioning

confidence: 99%

“…, i n ) ∈ N n , i 1 · · · i n } is a basis of both the n-fold symmetric injective tensor product⊗ n,s,ǫ X and the n-fold symmetric projective tensor product⊗ n,s,π X (see, e.g., [12]). However, they are not necessary unconditional bases (see, e.g., [14,18,20,8,17,6]).…”

Section: Introductionmentioning

confidence: 99%

Diagonals of injective tensor products of Banach lattices with bases

Lee

2016

Positivity

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Let E be a Banach lattice with a 1-unconditional basis {ei : i ∈ N}. Denote by ∆(⊗ n,ǫ E) (resp. ∆(⊗ n,s,ǫ E)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by ∆(⊗ n,|ǫ| E) (resp. ∆(⊗ n,s,|ǫ| E)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal {ei⊗· · ·⊗ei : i ∈ N} is a 1-unconditional basic sequence in both⊗ n,ǫ E and⊗ n,s,ǫ E. ⊗ n,s,ǫ X. In this paper, by using a result obtained in [2] we show that the tensor diagonal {e i ⊗ · · · ⊗ e i : i ∈ N} is a 1-unconditional basic sequence in⊗ n,s,ǫ X and we also show that the diagonal projections on both⊗ n,ǫ X and⊗ n,s,ǫ X are contractive.

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Unconditionality in Tensor Products and Ideals of Polynomials, Multilinear Forms and Operators

Cited by 7 publications

References 33 publications

The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

The symmetric Radon–Nikodým property for tensor norms

Diagonals of injective tensor products of Banach lattices with bases

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