-ideals of compact operators into
Kamil John, Dirk Werner (2000)
Czechoslovak Mathematical Journal
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We show for and subspaces of quotients of with a -unconditional finite-dimensional Schauder decomposition that is an -ideal in .
Kamil John, Dirk Werner (2000)
Czechoslovak Mathematical Journal
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We show for and subspaces of quotients of with a -unconditional finite-dimensional Schauder decomposition that is an -ideal in .
Paweł Barbarski, Rafał Filipów, Nikodem Mrożek, Piotr Szuca (2013)
Colloquium Mathematicae
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We consider the Katětov order between ideals of subsets of natural numbers ("") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).
B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)
Studia Mathematica
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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in , including one that has not been studied before. The proofs use various methods...
Vegard Lima, Åsvald Lima, Olav Nygaard (2004)
Studia Mathematica
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We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net of compact operators on X such that and in the strong operator topology. Similar results for dual spaces are also proved.
Pierre Matet (2013)
Fundamenta Mathematicae
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We discuss the problem of whether there exists a restriction of the noncofinal ideal on that is normal.
Ali Yassine, Mohammad Javad Nikmehr, Reza Nikandish (2024)
Czechoslovak Mathematical Journal
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Let be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of -ideals and a subclass of -absorbing primary ideals. A proper ideal of is called strongly 1-absorbing primary if for all nonunit elements such that , it is either or . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings over which every semi-primary ideal is strongly 1-absorbing primary, and rings over which...
Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)
Studia Mathematica
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Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on and for p < q. We also study the estimate above in the setting of Banach...
Hiroshi Sakai (2005)
Fundamenta Mathematicae
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We say that an ideal I on is semiproper if the corresponding poset is semiproper. In this paper we investigate properties of semiproper ideals on .
Olav Nygaard, Märt Põldvere (2009)
Studia Mathematica
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Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson’s projection P on (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with...
Khalid A. Mokbel (2016)
Mathematica Bohemica
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The concept of a -ideal in -distributive posets is introduced. Several properties of -ideals in -distributive posets are established. Further, the interrelationships between -ideals and -ideals in -distributive posets are investigated. Moreover, a characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of non-dense ideals. It is shown that every -ideal of a -distributive meet semilattice is semiprime. Several counterexamples are discussed. ...
Juan Manuel Delgado, Cándido Piñeiro (2013)
Studia Mathematica
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Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to for every Banach space Y and every T ∈ (Y,X), being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.
Manuel A. Fugarolas (2011)
Czechoslovak Mathematical Journal
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Let and . We prove that , the ideal of operators of Geľfand type , is contained in the ideal of -absolutely summing operators. For this generalizes a result of G. Bennett given for operators on a Hilbert space.
Khalid A. Mokbel (2015)
Mathematica Bohemica
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The concept of -ideals in posets is introduced. Several properties of -ideals in -distributive posets are studied. Characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal of a -distributive poset is non-dense, then is an -ideal. Moreover, it is shown that the set of all -ideals of a poset with forms a complete lattice. A result analogous to separation theorem for...
Stefania Gabelli (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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If is a domain with the ascending chain condition on (integral) invertible ideals, then the group of its invertible ideals is generated by the set of maximal invertible ideals. In this note we study some properties of and we prove that, if is a free group on , then is a locally factorial Krull domain.
Piotr Fijałkowski (2005)
Colloquium Mathematicae
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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form where is a continuous (2n+1)-linear operator.
Brian Lehmann (2014)
Annales de l’institut Fourier
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Given a pseudo-effective divisor we construct the diminished ideal , a “continuous” extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. Our main theorem shows that for most pseudo-effective divisors the multiplier ideal of the metric of minimal singularities on is contained in . We also characterize abundant divisors using the diminished ideal, indicating that the geometric and analytic information should coincide.
Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)
Archivum Mathematicum
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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if is a Banach space such that has a SRBF, then has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space has an approximative Schauder frame, then has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.
Ryan Causey (2013)
Studia Mathematica
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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index which is universal for the class of separable Banach spaces with Szlenk index not exceeding . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.
Volker Runde (2010)
Banach Center Publications
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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that . Still, ℬ(ℓ²) is...
Hermann Pfitzner (2010)
Bulletin of the Polish Academy of Sciences. Mathematics
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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains isometrically.