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Growth conditions on powers of Hermitian elements

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
University of Bristol
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Extract

Roe (7) has recently given a striking characterization of the sine function as essentially the only function defined on the real line which has all its derivatives and successive ‘integrals’ bounded by a uniform bound, and this result has since been generalized by Burkill (5). In this note we relate this work to Banach algebra theory in a natural way by demonstrating the equivalence of Theorems 1 and 2 below, and also by giving a direct proof of Theorem 1. (Theorem 2 is Burkill's generalization of Roe's result stated in complex form.) We enunciate a Banach space version of the result (Theorem 3), and a further generalization of all these results is given in Theorem 4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 1 , July 1982 , pp. 115 - 119
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Bollobás, B.A property of Hermitian elements. J. London Math. Soc. (2), 4 (1971), 379–380.Google Scholar
(2)Bollobás, B.The spatial numerical range and powers of an operator. J. London Math. Soc. (2), 7 (1973), 435–440.Google Scholar
(3)Bonsall, F. F. and Duncan, J.Numerical ranges of operators on normed spaces and of elements of normed algebras. London Math. Soc. Lecture Notes (Cambridge University Press, 1971).CrossRefGoogle Scholar
(4)Bonsall, F. F. and Duncan, J.Complete normed algebras (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
(5)Burkill, H.Sequences characterizing the sine function. Math. Proc. Cambridge Philos. Soc. 89 (1981), 71–77.CrossRefGoogle Scholar
(6)Kolmogorov, A. N.On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval (in Russian). Uzenye Moskovskogo Gosudarstvennego Mathematika 30 (1939), 3–13, Amer. Math. Soc. Transl. (1) 2 (1962), 233–243.Google Scholar
(7)Roe, J.A characterization of the sine function. Math. Proc. Cambridge Philos. Soc. 87 (1980), 69–73.CrossRefGoogle Scholar