On Certain Locally Convex Topologies on Banach Spaces

Jarchow, Hans

doi:10.1016/s0304-0208(08)71468-3

Functional Analysis: Surveys and Recent Results III, Proceedings of the Conference on Functional Analysis

1984

DOI: 10.1016/s0304-0208(08)71468-3

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On Certain Locally Convex Topologies on Banach Spaces

Hans Jarchow

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Cited by 4 publications

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“…It is known that S * (X * * , X * )| X = S * (X, X * ) (see [25]). Note that the Right and the Strong * topologies are particular cases of the topologies defined in [17].…”

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confidence: 99%

“…[25,Example 3.19]). Note that there is an "overlap", in the case of the Right topology, between [24,25,22] and [17,3,11,12]. Indeed, in [12] (see also [13]), starting from the work done in [3] and [11], the authors proved the more complete result below, relating the above kind of continuity to the uniform Right × • • • × Right-to-norm continuity on bounded sets.…”

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confidence: 99%

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Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples

2009

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In recent papers, the Right and the Strong * topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong *-to-norm continuity of a multilinear operator T defined on C *-algebras (respectively, JB *-triples) to C *-summability (respectively, JB *-triple-summability).

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“…It is known that S * (X * * , X * )| X = S * (X, X * ) (see [25]). Note that the Right and the Strong * topologies are particular cases of the topologies defined in [17].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples

2009

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Remarks on a characterization of nuclearity

Jarchow

1984

Arch. Math

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Injective factorization of holomorphic mappings

Gutiérrez¹

1999

Proc. Amer. Math. Soc.

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Abstract. We characterize the holomorphic mappings f between complex Banach spaces that may be written in the form f = g • T , where g is another holomorphic mapping and T is an operator belonging to a closed injective operator ideal. Analogous results are previously obtained for multilinear mappings and polynomials.In recent years, several authors [1], [6], [7] have studied conditions on a holomorphic mapping f between complex Banach spaces so that it may be written in the form f = g • T , where g is another holomorphic mapping and T is a (linear bounded) operator belonging to certain classes of operators.In this paper, we look at this problem in the setting of operator ideals, thus finding more general conditions so that these factorizations occur. Our results include all the previous ones, with simpler proofs, and apply to many new cases.A linear mapping T belongs to the ideal Co of compact operators if and only if it is weakly (uniformly) continuous on bounded subsets. So, if a mapping f : E → F may be written as f = g • T with T ∈ Co, then f is weakly uniformly continuous on bounded subsets of E (the mappings with this property have been studied by numerous authors, for instance [2], [3], [5]). The authors have shown in [7] that the converse also holds: if a holomorphic mapping f between Banach spaces is weakly uniformly continuous on bounded subsets, then it can be factorized in the form f = g • T , with T ∈ Co. A similar result was proved in [1] for T in the ideal WCo of weakly compact operators, and F = C the complex field.In [6, Satz 2.1], using interpolation techniques, the factorization f = g • T has been characterized in terms of the derivatives of f , for T in any closed, injective and surjective operator ideal, and F = C.In this paper, for any closed injective operator ideal U, we find several conditions on a polynomial P : E → F (or a multilinear map) so that it may be written as P = Q • T , with T in U. As a consequence, we prove that a holomorphic map f : E → F admits such a factorization if and only if f is uniformly continuous on bounded sets with respect to a suitable topology τ U . In the case U = Co, the topology τ U is the finest l.c. topology that coincides with the weak topology on bounded sets. Consequently, using a simpler proof, we recover the result in [7].

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On Certain Locally Convex Topologies on Banach Spaces

Cited by 4 publications

References 16 publications

Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples

Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples

Remarks on a characterization of nuclearity

Injective factorization of holomorphic mappings

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On Certain Locally Convex Topologies on Banach Spaces

Cited by 4 publications

References 16 publications

Factorizing multilinear operators on Banach spaces, C*-algebras and JB*-triples

Factorizing multilinear operators on Banach spaces, C*-algebras and JB*-triples

Remarks on a characterization of nuclearity

Injective factorization of holomorphic mappings

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Product

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Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples

Factorizing multilinear operators on Banach spaces, C^-algebras and JB^-triples