Lower bounds for projective designs, cubature formulas and related isometric embeddings

Lyubich, Yu. I.

doi:10.1016/j.ejc.2008.08.001

Cited by 6 publications

(5 citation statements)

References 25 publications

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“…In a recent article [6], the authors have studied isometric embeddability of S m q into S n p for 1 ≤ p = q ≤ ∞ and m, n ≥ 2. This partially established a non-commutative analogue of the results appearing in [26], [27], [28], [29], [30]. The motivation behind this study was two-fold.…”

Section: Introduction and Main Resultssupporting

confidence: 55%

“…We refer [16], [17], [18], [15], [43], [19], [37], [11], [20] and references therein for more information in this direction of research. Therefore, in view of the results appearing in [26], [27], [28], [29], [30], it is natural to study non-commutative analogs. The second motivation behind [6] was again operator space theory and its connections to boundary normal dilation, where such non-commutative isometric embeddability was crucial.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

Chattopadhyay¹,

Hong²,

Pradhan³

et al. 2022

Preprint

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The existence of isometric embedding of S m q into S n p , where 1 ≤ p = q ≤ ∞ and m, n ≥ 2 has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above mentioned range of p and q. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space ℓ m q (R) into ℓ n p (R), where (q, p) ∈ (0, ∞) × (0, 1) and p = q. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ (0, 2) × (0, 1) ∪ {1} × (0, 1) \ { 1 n : n ∈ N} ∪ {∞} × (0, 1) \ { 1 n : n ∈ N} and p = q. Moreover, in some restrictive cases, we also show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ [2, ∞) × (0, 1). To achieve our goal we significantly use Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

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Section: Introduction and Main Resultssupporting

confidence: 55%

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

Chattopadhyay¹,

Hong²,

Pradhan³

et al. 2022

Preprint

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“…For rank-1 symmetric spaces or 1-transitive distance regular graphs, λ(Ω) and V (Ω) can be related by the behavior of the first roots of the corresponding family of orthogonal polynomials. Plugging this into Theorem 1.2, we recover the first linear programming bound on 1-transitive distance regular graphs [12] and on rank-1 symmetric spaces ( [19] for the sphere, and [15] for projective spaces).…”

Section: Discussionmentioning

confidence: 85%

Lower bounds for designs in symmetric spaces

Eidelstein¹,

Samorodnitsky²

2010

Preprint

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A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube.We prove lower bounds on designs in spaces with a large group of symmetries. These spaces include globally symmetric Riemannian spaces (of any rank) and commutative association schemes with 1-transitive group of symmetries.Our bounds are, in general, implicit, relying on estimates on the spectral behavior of certain symmetry-invariant linear operators. They reduce to the first linear programming bound for designs in globally symmetric Riemannian spaces of rank-1 or in distance regular graphs. The proofs are different though, coming from viewpoint of abstract harmonic analysis in symmetric spaces. As a dividend we obtain the following geometric fact: a design is large because a union of "spherical caps" around its points "covers" the whole space.

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“…This topic is also closely related to Warning's problem, cubature formulae and spherical design. We refer [17][18][19]21] and [20] for more information in this direction.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…These tools enable us to obtain new isometry from existing isometry and finally use classic trick of 'infinite descent' (see [8] for many other illustrations) to obtain contradiction. Our approach is even new in the commutative case and we believe that our methods can be applied to study isometries between other kind of Banach spaces also, for example, Orlicz spaces and symmetric operator spaces and might be helpful in simplifying existing proofs in [17][18][19][20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On isometric embedding ℓpm→S∞ and unique operator space structure

Ray

2020

Bull. London Math. Soc.

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We study existence of linear isometric embedding of m p into S∞, for 1 p < ∞, and unique operator space structure on two-dimensional Banach spaces. For p ∈ (2, ∞) ∪ {1}, we show that indeed 2 p does not embed isometrically into S∞. This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza (Houston J. Math. 44 (2018) 1205-1212. We also show that S m 1 does not embed isometrically into S n p for all 1 < p < ∞ and m 2. As a consequence, we establish non-commutative analogue of some of the results of Lyubich and Shatalova (Algebra i Analiz 16 (2004) 15-32). We also show that (C 2 , . Bp,q ) does not embed isometrically into S∞ for 2 < p, q < ∞. The main ingredients in our proofs are notions of Birkhoff-James orthogonality and norm-parallelism for operators on Hilbert spaces. These enable us to deploy 'infinite descent' type of arguments to obtain contradictions. Our approach is new even in the commutative case. We prove that (C 2 , . Bp,q ) does not have unique operator space structure whenever (p, q)∞) by showing that they do not have Property P or two summing property. In view of Misra, Pal and Varughese (J. Operator Theory 82 (2019) 23-47), this produces genuinely new examples of two-dimensional Banach spaces without unique operator space structure, providing a partial answer to a question of Paulsen. In this case, we derive our result by transferring the problem to real case and applying known results of Arias, Figiel, Johnson and Schechtman (Trans.

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Lower bounds for projective designs, cubature formulas and related isometric embeddings

Cited by 6 publications

References 25 publications

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

Lower bounds for designs in symmetric spaces

On isometric embedding ℓpm→S∞ and unique operator space structure

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