Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra. (English) Zbl 1172.47002

The author gives a characterization of the extended eigenvalues and eigenvectors of the integral operator \(Vf(z)= \int_0^z f(t)\,dt\) in the Wiener algebra \(\{f(z)=\sum_{n=0}^\infty a_nz^n: \sum_{n=0}^\infty |a_n| <\infty\}\), that is, those complex numbers \(\lambda\) and bounded operators \(A\) on the Wiener algebra such that \(VA=\lambda AV\). A similar result for some weighted shift operators on \(\ell_p\) spaces is given as well.


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