2014
DOI: 10.2478/s11533-014-0423-0
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Composition results for strongly summing and dominated multilinear operators

Abstract: Abstract:In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible -linear versions of Grothendieck's composition theorem in the case ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact. MSC:47H60, 46G25, 46B25,… Show more

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Cited by 9 publications
(15 citation statements)
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References 34 publications
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“…The notion of operator ideals was systematized by Pietsch [30] and multi-ideals were introduced by Pietsch [31] and has been developed by several authors (for recent contributions, see, e.g., [14,22,27,28,32,35]). Definition 2.5.…”
Section: Proof (A) Consider the Continuous Linear Operatormentioning
confidence: 99%
See 1 more Smart Citation
“…The notion of operator ideals was systematized by Pietsch [30] and multi-ideals were introduced by Pietsch [31] and has been developed by several authors (for recent contributions, see, e.g., [14,22,27,28,32,35]). Definition 2.5.…”
Section: Proof (A) Consider the Continuous Linear Operatormentioning
confidence: 99%
“…It was introduced by Pietsch [31] and its theory has been developed by several authors, e.g. [10,16,23,32]. Considering the linearly stable sequence classes ℓ p (·) and ℓ w p j (·), as ℓ p 1 · · · ℓ pn 1 ֒→ ℓ p , we have L d;p 1 ,...,pn = L ℓ w p 1 (·),...,ℓ w pn (·);ℓp(·) .…”
Section: This Givesmentioning
confidence: 99%
“…The theory of ideals of multilinear operators between Banach spaces (multi-ideals) was initiated by A. Pietsch [30] as a first step to take to the nonlinear setting the successful theory of ideals of linear operators (operator ideals). Since then much research has been done in this subject, we mention just a few recent developments: [1,3,4,5,6,7,14,15,16,19,27,31,32,33,35].…”
Section: Introductionmentioning
confidence: 99%
“…Se pensarmos que estamos tratando com aplicações multilineares, istoé, com aplicações n-lineares com n arbitrário, e não com aplicações n-lineares com n fixo, faz todo sentido compor com aplicações multilinearesà esquerda e obter a composta com grau de multilinearidade maior que a aplicação original. Faz sentido também estudar classes que são estáveis em relação a esse tipo de composição, e isso já foi objeto de estudo de alguns especialistas, veja, por exemplo, [23,46,47]. O ob-jetivo central desta teseé explorar, em sua totalidade, a natureza multilinear do assunto, e para isso introduzimos e desenvolvemos a teoria de classes de aplicações multilineares que gozam deste tipo de estabilidade; as quais optamos por chamar de hiper-ideais de aplicações multilineares.…”
Section: Definição E Propriedades Geraisunclassified
“…, B n ) também pertenceà classe H. Acreditamos que, dessa forma, estamos explorando de forma mais aprofundada a natureza multilinear da situação. A condição de hiper-ideal já foi estudada em alguns poucos casos particulares, por exemplo em [23,47]. O objetivo desta teseé introduzir e desenvolver de forma sistemática a teoria de hiper-ideais de aplicações multilineares entre espaços de Banach.…”
Section: Introductionunclassified