Functional equations with shift in spaces of analytic functions in a halfplane. (English. Russian original) Zbl 0516.47017

Ukr. Math. J. 34, 543-547 (1983); translation from Ukr. Mat. Zh. 34, No. 5, 661-666 (1982).
The author describes the spectra of two operators: \[ (A_1f)(t)=g(t)f(t+i\beta),\quad \beta>0, \] and \[ (A_2f)(t)=g(t)f(\alpha t+i\beta),\quad\alpha>0, \alpha\neq 1, \beta>0 \] in the Hardy spaces \(H_p\), \(1<p<\infty\).
Theorem 2: The spectrum of \(A_1\) in \(H_p\) is the set \(\sigma=:\{z: z=re^{t\arg a}, 0\leq r\leq a\}\), where \(a\) is the constant in the representation of \(g(t)\) in the form \(g(t)=a+g_0(t)\) (\(g_0(t)\in H_0^c\)). If \(\lambda_0\in\sigma\) then, in the sense of Kreĭn, \(A_1-\lambda_0I\) is neither \(n\)-normal nor \(d\)-normal in \(H_p\).
Theorem 4. The spectrum of the operator \(A_2\) in \(H_p\) coincides with the disk of radius \(| a| \alpha^{-1/p}\) where \(a\) is the constant as above. For every \(\lambda_0\in\sigma_p\) the operator \(A_2 -\lambda_0I\) is neither \(n\)- nor \(d\)-normal.
(revised version)
Reviewer: V. A. Tkachenko


47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47A10 Spectrum, resolvent
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[1] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall (1962). · Zbl 0117.34001
[2] A. P. Calderon, ?Intermediate spaces and interpolation; the complex method,? Stud. Math.,24, 113-120 (1964).
[3] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press (1971). · Zbl 0232.42007
[4] N. Dunford and J. T. Schwartz, Linear Operators, General Theory, Vol. 1, Wiley-Interscience, New York (1958).
[5] A. Zymund, Trigonometrical Series, Cambridge University Press (1977).
[6] S. G. Krein, Linear Differential Equations in Banach Space, Am. Math. Soc. (1972). · Zbl 0236.47035
[7] G. S. Litvinchuk, Boundary-Value Problems and Singular Equations with Shift [in Russian], Nauka, Moscow (1977). · Zbl 0462.30029
[8] V. V. Shevchik, ?Functional equations with shift in spaces of analytic functions in a halfplane,? Dokl. Akad. Nauk SSSR,256, 51-53 (1981). · Zbl 0469.30025
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