Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 570783
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B05, 03H05

Retrieve articles in all journals with MSC: 47B05, 03H05

Additional Information

Keywords: Banach space, Hilbert space, bounded linear transformation, nonstandard hull
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society