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A characterization of the circle and its applications to Hermitian operators

Published online by Cambridge University Press:  24 October 2008

Béla Bollobás
Béla Bollobás
Affiliation:
University of Cambridge
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Extract

Let X be a complex normed space and let T be a Hermitian linear operator on X (i.e. the numerical range of T is real). It was proved by Bonsall (see (4); Theorem 10.13) that then

for every u ∈ X. This inequality was improved by the author (3) to

Inequality (1) is closely related (see (3)) to some inequalities of Hadamard (6) and Kolmogorov (7) about the successive derivatives of functions in L∞(−∞, ∞). It was also shown in (3) (and was, in fact, shown already in (7)) that the constant 2 is best possible in (1). However, as we shall see, inequality (1) can be sharpened considerably if T attains its norm on u, i.e. if ‖Tu‖ = ‖T‖‖u‖.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 69 - 72
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

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