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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonstandard extensions of transformations between Banach spaces


Author: D. G. Tacon
Journal: Trans. Amer. Math. Soc. 260 (1980), 147-158
MSC: Primary 47B05; Secondary 03H05
DOI: https://doi.org/10.1090/S0002-9947-1980-0570783-0
MathSciNet review: 570783
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Abstract: Let X and Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an $ {\aleph _1}$-saturated enlargement by $ \hat X$ and $ \hat Y$ respectively. If $ {\mathcal{B}}\,(X,\,Y)$ denotes the space of bounded linear transformations then a subset S of elements of $ {\mathcal{B}}\,(X,\,Y)$ extends naturally to a subset $ \hat S$ of $ {\mathcal{B}}\,(\hat X,\,\hat Y)$. This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that $ ({\mathcal{B}}\,(X,\,Y)\hat )\,\mathop \subset \limits_ \ne \,{\mathcal{B}}\,(\hat X,\,\hat Y)$ provided X and Y are infinite dimensional and contrasts this with the inclusion $ {\mathcal{K}}(\hat H)\,\mathop \subset \limits_ \ne \,({\mathcal{K}}(H)\hat )$ where $ {\mathcal{K}}(H)$ denotes the space of compact operators on a Hilbert space.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0570783-0
Keywords: Banach space, Hilbert space, bounded linear transformation, nonstandard hull
Article copyright: © Copyright 1980 American Mathematical Society