Functional equations with shift in spaces of analytic functions in a halfplane (Q1052614)

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)