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On a summability theorem of Berg, Crawford and Whitley

Published online by Cambridge University Press:  24 October 2008

D. J. H. Garling  and
A. Wilansky
D. J. H. Garling
Affiliation:
St John's College, Cambridge and Lehigh University, Bethlehem, Pennsylvania
A. Wilansky
Affiliation:
St John's College, Cambridge and Lehigh University, Bethlehem, Pennsylvania
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Extract

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:

Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 3 , May 1972 , pp. 495 - 497
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Banach, S.Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
(2)Berg, I. D.A Banach algebra criterion for Tauberian theorems. Proc. Amer. Math. Soc. 15 (1964), 648–652.CrossRefGoogle Scholar
(3)Crawford, J. P. Transformations in Banach spaces, Lehigh University dissertation, 1966.Google Scholar
(4)Dunford, N. and Schwartz, J. T.Linear operators, Part 1 (Interscience, 1958).Google Scholar
(5)Köthe, G.Topological Vector Spaces I (Springer Verlag, 1969).Google Scholar
(6)Wilansky, A.Functional analysis (Blaisdell Publishing Company, 1964).Google Scholar
(7)Wilansky, A.Topics in functional analysis (Springer Verlag, 1967).CrossRefGoogle Scholar
(8)Wilansky, A.Topological divisors of zero and Tauberian theorems, Trans. Amer. Math. Soc. 113 (1964), 240–251.CrossRefGoogle Scholar
(9)Whitley, R. J.Conull and other matrices which sum a bounded divergent sequence Amer. Math. Monthly 74 (1967), 798–801.CrossRefGoogle Scholar
(10)Zeller, K.Faktorfolgen bei Limitierungsverfahren, Math. Z. 56 (1952), 134–151.CrossRefGoogle Scholar