×

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.

MSC:

47A10 Spectrum, resolvent
47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biswas, A.; Lambert, A.; Petrovic, S., Extended eigenvalues and the Volterra operator, Glasgow math. J., 44, 521-534, (2002) · Zbl 1037.47013
[2] Brown, S., Connections between an operator and a compact operator that yield hyperinvariant subspaces, J. oper. theory, 1, 117-121, (1979) · Zbl 0451.47002
[3] Karaev, M.T., Application of the Duhamel product in problems of the basis property, Izv. akad. nauk az. SSR ser. fiz. tekh. mat. nauk, 3-6, 145-150, (1990), (in Russian)
[4] Karaev, M.T., On extended eigenvalues and extended eigenvectors of some operator classes, Proc. am. math. soc., 134, 2383-2392, (2006) · Zbl 1165.47302
[5] Karaev, M.T.; Mustafayev, H.S., On some properties of deddens algebras, Rocky mount. J. math., 33, 915-926, (2003) · Zbl 1079.46032
[6] Karaev, M.T.; Saltan, S., A Banach algebra structure for the Wiener algebra \(W(\mathbb{D})\) of the disc, Complex variables theory appl., 50, 299-305, (2005) · Zbl 1082.47028
[7] Lambert, A.; Petrovic, S., Beyond hyperinvariance for compact operators, J. funct. anal., 219, 93-108, (2005) · Zbl 1061.47018
[8] Lomonosov, V., Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. anal. i prilozhen., 7, 3, 55-56, (1973), (in Russian)
[9] C. Pearcy, Some recent developments in operator theory, in: Regional Conference Series in Mathematics, vol. 36, American Mathematical Society, Providence, RI, 1978. · Zbl 0444.47001
[10] Roth, P.G., Bounded orbits of conjugation, analytic theory, Indiana univ. math. J., 32, 4, 491-509, (1983) · Zbl 0495.47008
[11] Shkarin, S., Compact operators without extended eigenvalues, J. math. anal. appl., 332, 455-462, (2007) · Zbl 1121.47012
[12] Wigley, N.M., The Duhamel product of analytic functions, Duke math. J., 41, 211-217, (1974) · Zbl 0283.30036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.