Nonstandard extensions of transformations between Banach spaces

Tacon, D. G.

doi:10.1090/S0002-9947-1980-0570783-0

Nonstandard extensions of transformations between Banach spaces
HTML articles powered by AMS MathViewer

by D. G. Tacon PDF

Trans. Amer. Math. Soc. 260 (1980), 147-158 Request permission

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 $\widehat {\mathcal {B}(X,Y)} \subsetneqq \mathcal {B}(\hat {X}, \hat {Y})$ provided $X$ and $Y$ are infinite dimensional and contrasts this with the inclusion $\mathcal {K}(\hat H) \subsetneqq \widehat {\mathcal {K}(H)}$ where $\mathcal {K}(H)$ denotes the space of compact operators on a Hilbert space.

References

P. M. Anselone and T. W. Palmer, Spectral analysis of collectively compact, strongly convergent operator sequences, Pacific J. Math. 25 (1968), 423–431. MR 227807
A. L. Brown and A. Page, Elements of functional analysis, The New University Mathematics Series, Van Nostrand Reinhold Co., London-New York-Toronto, Ont., 1970. MR 0358266
S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 9, Marcel Dekker, Inc., New York, 1974. MR 0415345
David Cozart and L. C. Moore Jr., The nonstandard hull of a normed Riesz space, Duke Math. J. 41 (1974), 263–275. MR 358281

Linear operators

C. Ward Henson, When do two Banach spaces have isometrically isomorphic nonstandard hulls?, Israel J. Math. 22 (1975), no. 1, 57–67. MR 385525, DOI 10.1007/BF02757274
C. Ward Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), no. 1-2, 108–144. MR 461104, DOI 10.1007/BF02756565
C. Ward Henson and L. C. Moore Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405–435. MR 308722, DOI 10.1090/S0002-9947-1972-0308722-5
C. Ward Henson and L. C. Moore Jr., Subspaces of the nonstandard hull of a normed space, Trans. Amer. Math. Soc. 197 (1974), 131–143. MR 365098, DOI 10.1090/S0002-9947-1974-0365098-7
C. Ward Henson and L. C. Moore Jr., Nonstandard hulls of the classical Banach spaces, Duke Math. J. 41 (1974), 277–284. MR 367625
Robert C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. MR 165344, DOI 10.1090/S0002-9947-1964-0165344-2
Robert C. James, Super-reflexive Banach spaces, Canadian J. Math. 24 (1972), 896–904. MR 320713, DOI 10.4153/CJM-1972-089-7
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
L. C. Moore Jr., Hyperfinite extensions of bounded operators on a separable Hilbert space, Trans. Amer. Math. Soc. 218 (1976), 285–295. MR 402524, DOI 10.1090/S0002-9947-1976-0402524-0
Albrecht Pietsch, Nuclear locally convex spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66, Springer-Verlag, New York-Heidelberg, 1972. Translated from the second German edition by William H. Ruckle. MR 0350360
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964), 252–261. MR 177302, DOI 10.1090/S0002-9947-1964-0177302-2