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Matrices and norms

Published online by Cambridge University Press:  24 October 2008

G. M. Petersen
G. M. Petersen
Affiliation:
The University of New Mexico and University College, Swansea
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Extract

We shall define the norm h(A) of a regular summability matrix A = (amn) by Two matrices are said to be b-equivalent if every bounded sequence summable by on matrix is summable by the other. If A sums all bounded sequences that are summable by B, A is said to be b-stronger than B. The norm of a method is defined as , where the inf is taken over all the matrices equivalent to for bounded sequences. These norms have been investigated by Brudno(1). One of his main results is that, if sums all bounded sequences that are summable, then In paper we shall prove the following.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 2 , April 1961 , pp. 271 - 273
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Brudno, A., Summation of bounded sequences by matrices. Mat. Sbornik, 16 (1945), 191–247 (in Russian).Google Scholar
(2)Petersen, G. M., Matrix norms. Quart. J. Math. (2), 9 (Oxford, 1958), 161–8.CrossRefGoogle Scholar
(3)Petersen, G. M., Norms of summation methods. Proc. Camb. Phil. Soc. 54 (1958), 354–7.CrossRefGoogle Scholar