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On the spectrum of an integral operator

Published online by Cambridge University Press:  18 May 2009

P. G. Rooney
P. G. Rooney
Affiliation:
University of Toronto, Toronto, Ontario, Canada, M5S 1A1
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The integral operator which we will consider in this paper is the operator T denned for suitably restricted functions f on (0, ∞) by

where x >0 and the integral is taken in the Cauchy principal value sense at t = x. This operator plays a considerable role in Wiener–Hopf theory; see [2; Chapter 5].

Since T is clearly the restriction to (0, ∞) of minus the Hilbert transformation applied to functions which vanish on (−∞, 0), it follows easily from the theory of the Hilbert transformation, as given in say [6; Theorem 101], that T is a bounded operator from Lp(0, ∞) to itself for 1 < p < ∞.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 5 - 9
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

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