## 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

### MSC:

 47B38 Linear operators on function spaces (general) 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 47A10 Spectrum, resolvent

### Keywords:

Hardy space; functional equations with shift
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### References:

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