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Hilbert transforms and unitary equivalence

Published online by Cambridge University Press:  18 May 2009

R. Putnam
R. Putnam
Affiliation:
Purdue UniversityLafayette, Indiana
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If E is a subset of the real line of positive measure, then the associated Hilbert transform H = HE,

where the integral is a Cauchy principal value, is a bounded self-adjoint operator on L2(E) (cf. Muskhelishvili [4]). In case E = (-∞, ∞) the transformation is also unitary with a spectrum consisting of 1 and -1, each of infinite multiplicity (Titchmarsh [10]). If E is a inite interval the spectral representation of H has been given by Koppelman and Pincus [3]; see also Putnam [6]. In particular the spectrum of H is in this case the closed interval [-1, 1]. Moreover, according to Widom [11], the spectrum of H is [-1, 1] whenever E ≠ (-∞, ∞), that is, whenever

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 8 , Issue 2 , July 1967 , pp. 113 - 117
Copyright
Copyright © Glasgow Mathematical Journal Trust 1967

References

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