Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T02:02:40.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Inclusions between FK spaces and Kuttner's theorem

Published online by Cambridge University Press:  24 October 2008

I. J. Maddox
I. J. Maddox
Affiliation:
The Queen's University of Belfast
Get access Rights & Permissions [Opens in a new window]

Extract

The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesàro summable with index p but which is not A summable. This theorem was extended by Maddox[3] to coregular matrices, and Thorpe [9] gave a further extension by showing that if 0 < p < 1 and X is a locally convex FK space with X ⊃ w0(p) then X ⊃ l∞. Here, w0(p) denotes the space of sequences strongly summable to 0, i.e. x∈w0(p) if and only if

and l∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 523 - 527
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Cooke, R. G.. Infinite Matrices and Sequence Spaces (Macmillan, 1950).Google Scholar
[2]Kuttner, B.. Note on strong summability. J. London Math. Soc. 21 (1946), 118–122.Google Scholar
[3]Maddox, I. J.. On Kuttner's theorem. J. London Math. Soc. 43 (1968), 285–290.CrossRefGoogle Scholar
[4]Maddox, I. J.. FK spaces which include strongly summable sequences. Math. Proc. Cambridge Philos. Soc. 93 (1983), 131–134.CrossRefGoogle Scholar
[5]Maddox, I. J.. Series in locally convex spaces and inclusions between FK spaces. Math. Proc. Cambridge Philos. Soc. 95 (1984), 467–472.CrossRefGoogle Scholar
[6]Maddox, I. J.. Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1986), 161–166.CrossRefGoogle Scholar
[7]Ruckle, W. H.. FK spaces in which the sequence of coordinate vectors is bounded. Canad. J. Math. 25 (1973), 973–978.CrossRefGoogle Scholar
[8]Snyder, A. K. and Wilansky, A.. Inclusion theorems and semiconservative FK spaces. Rocky Mountain J. Math. 2 (1972), 595–603.CrossRefGoogle Scholar
[9]Thorpe, B.. An extension of Kuttner's theorem. Bull. London Math. Soc. 13 (1981), 301–302.CrossRefGoogle Scholar
[10]Zeller, K.. Allgemeine Eigenschaften von Limitierungsverfahren. Math. Zeit. 53 (1951), 463–487.CrossRefGoogle Scholar