Geodesic
Hilbert transforms and unitary equivalence
Glasgow mathematical journal, Tome 8 (1967) no. 2, pp. 113-117

Voir la notice de l'article provenant de la source Cambridge University Press

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 
Putnam, R. Hilbert transforms and unitary equivalence. Glasgow mathematical journal, Tome 8 (1967) no. 2, pp. 113-117. doi: 10.1017/S0017089500000161
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[1] 1.Hopf, E., Ergodentheorie, Chelsea (New York, 1948). Google Scholar

[2] 2.Kato, T., Perturbation of continuous spectra by trace class operators, Proc. Japan Acad. 33 (1957), 260–264. Google Scholar

[3] 3.Koppelman, W. and Pincus, J., Spectral representations for finite Hilbert transformations, Math. Zeit. 71 (1959), 399–407. Google Scholar | DOI

[4] 4.Muskhelishvili, N. J., Singular integral equations (Groningen, 1953). Google Scholar

[5] 5.Putnam, C. R., On differences of unitarily equivalent self-adjoint operators, Proc. Glasgow Math. Assoc. 4 (1960), 103–107. Google Scholar | DOI

[6] 6.Putnam, C. R., The spectra of generalized Hilbert transforms, /. Math. Mech. 14 (1965), 857–872. Google Scholar

[7] 7.Riesz, F. and Sz.-Nagy, B., Functional analysis, Ungar (New York, 1955). Google Scholar

[8] 8.Rosenblum, M., Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math. 7 (1957), 997–1010. Google Scholar | DOI

[9] 9.Schatten, R., A theory of cross spaces, Ann. Math. Studies (Princeton, 1950). Google Scholar

[10] 10.Titchmarsh, E. C., Introduction to the theory of Fourier integrals (Oxford, 1948). Google Scholar

[11] 11.Widom, H., Singular integral equations in L, Trans. Amer. Math. Soc. 97 (1960), 131–160. Google Scholar

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