Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients

Kwapień, Stanisław

doi:10.4064/sm-44-6-583-595

Cited by 278 publications

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“…The trivial estimate Ff ∞ (1/ √ 2π) f 1 shows that each Banach space has Fourier type 1. A Banach space has Fourier type 2 if and only if it is isomorphic to a Hilbert space [15] (see also [16, pp. 73, 74]).…”

Section: Proof Assume That Each Sequence (mentioning

confidence: 99%

Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Arendt¹,

Bu²

2004

Proceedings of the Edinburgh Mathematical Society

127

215

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Let 1 p, q ∞, s ∈ R and let X be a Banach space. We show that the analogue of Marcinkiewicz's Fourier multiplier theorem on L p (T) holds for the Besov space B s p,q (T; X) if and only if 1 < p < ∞ and X is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann's result (Math. Nachr. 186 (1997), 5-56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

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Section: Proof Assume That Each Sequence (mentioning

confidence: 99%

Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Arendt¹,

Bu²

2004

Proceedings of the Edinburgh Mathematical Society

127

215

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“…As a consequence it follows that Pruitt's estimates hold precisely when B is both type 2 and cotype 2, that is by Kwapien's theorem [11], when B is isomorphic to a Hilbert space. We observe in passing that since R d with the usual norm can be isometrically embedded in the Hilbert space ℓ 2 , this implies the constants in (1.2) and (1.3) may in fact be taken independent of dimension.…”

Section: Theorem 12 the Following Are Equivalentmentioning

confidence: 93%

Pruitt’s Estimates in Banach Space

Griffin

2009

J Theor Probab

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“…Un espace de Hilbert està la fois de type 2 et de cotype 2, et un résultat profond de Kwapień [10] montre que réciproquement les seuls espaces de Banach qui sontà la fois de type 2 et de cotype 2 sont les espaces isomorphes aux espaces de Hilbert.…”

Section: Type Et Cotype Des Espaces De Banachunclassified

Sur quelques extensions au cadre banachique de la notion d'opérateur de Hilbert–Schmidt

Abdillah¹,

Esterlé²,

Haak³

2015

Studia Math.

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Résumé. -Le but de cet article est de faire le point sur diverses possibilités connues pouŕ etendre au cadre Banachique la notion d'opérateur de Hilbert-Schmidt : opérateurs p-sommants, γ-sommants ou γ-radonifiants, opérateurs faiblement * 1-nucléaires et classes d'opérateurs définies par des propriétés de factorisation. On introduit la classe PS 2 (E; F ) des opérateurs pré-HilbertSchmidt commeétant la classe des opérateurs u : E → F tels que w • u • v soit Hilbert-Schmidt pour tout opérateur borné v : H 1 → E et tout opérateur borné w : F → H 2 , H 1 et H 2 désignant des espaces de Hilbert quelconque. Hormis le cas trivial où l'un des deux espaces E ou F est un "espace de Hilbert-Schmidt", cet espace ne semble avoirété décrit que dans le cas banal où l'un des deux espaces E et F est un espace de Hilbert.Mots clés : Espaces de Banach, opérateurs de Hilbert-Schmidt, opérateurs p-sommants, opérateurs presque sommants, opérateurs γ-sommants, opérateurs γ-radonifiants, inégalité de Grothendieck.Abstract : In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators : p-summing operators, γ-summing or γ-radonifying operators, weakly * 1-nuclear operators and classes of operators defined via factorization properties. We introduce the class PS 2 (E; F ) of pre-Hilbert-Schmidt operators as the class of all operators u : E → F such that w • u • v is Hilbert-Schmidt for every bounded operator v : H 1 → E and every bounded operator w : F → H 2 , where H 1 et H 2 are Hilbert spaces. Besides the trivial case where one of the spaces E or F is a "Hilbert-Schmidt space", this space seems to have been described only in the easy situation where one of the spaces E or F is a Hilbert space.

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Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients

Cited by 278 publications

References 0 publications

Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Operator-Valued Fourier Multipliers on Periodic Besov Spaces and Applications

Pruitt’s Estimates in Banach Space

Sur quelques extensions au cadre banachique de la notion d'opérateur de Hilbert–Schmidt

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