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Compactness in topological tensor products and operator spaces

Name: Compactness in topological tensor products and operator spaces
Creator: J. R. Holub
Keywords: Normed linear spaces and Banach spaces; Banach lattices, Dual spaces of operator algebras, Tensor products in functional analysis, 0101 mathematics, 10. No inequality, 16. Peace & justice, 01 natural sciences, Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

descriptionPublicationkeyboard_double_arrow_right Article , Other literature type 01 Feb 1972Publisher:American Mathematical Society (AMS)Journal:Proceedings of the American Mathematical Society, volume 36, pages 398-398 (issn: 0002-9939,

Authors: J. R. Holub;

doi: 10.1090/s0002-9939-1972-0326458-7 , 10.2307/2039168

Compactness in topological tensor products and operator spaces

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Abstract

Let E and F be Banach spaces, E(?F their algebraic tensor product, and E (?,a F the completion of E(?F with respect to a uniform crossnorm oc?i (where ) is the "least", and y the greatest, crossnorm). In ?2 we characterize the relatively compact subsets of E (DA Fas those which, considered as spaces of operators from E* to F and from F* to E, take the unit balls in E* and in F* to relatively compact sets in F and E, respectively. In ?3 we prove that if T1: E1-E2 and T2: F1--F2 are compact operators then T1 (DA T2 and T1 ?J T2 are each compact, and results concerning the problem for an arbitrary crossnorm oc are also given. Schatten has characterized (E (D F) * as a certain space of operators of "finite oc-norm". In ?4 we show that a space of operators has such a representation if and only if its unit ball is weak operator compact.

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Virginia Tech
United States

Keywords

Normed linear spaces and Banach spaces; Banach lattices, Dual spaces of operator algebras, Tensor products in functional analysis, Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

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