Geodesic
Connections between Deddens Algebras and Extended Eigenvectors
Matematičeskie zametki, Tome 90 (2011) no. 1, pp. 40-44.

Voir la notice de l'article provenant de la source Math-Net.Ru

A complex number $\lambda$ is called an extended eigenvalue of the shift operator $S$, $Sf=zf$, on the disc algebra $C_{A}(\mathbb{D})$ if there exists a nonzero operator $A\colon C_{A}(\mathbb{D}) \to C_{A}(\mathbb{D})$ satisfying the equation $AS=\lambda S\mspace{-3mu}A$. We describe the set of all extended eigenvectors of $S$ in terms of multiplication operators and composition operators. It is shown that there are connections between the Deddens algebra associated with $S$ and the extended eigenvectors of $S$.
Keywords: disc algebra, multiplication operator, extended eigenvalue, extended eigenvector, shift operator, Banach algebra.
Mots-clés : Deddens algebra 
@article{MZM_2011_90_1_a4,
     author = {M. Gurdal},
     title = {Connections between {Deddens} {Algebras} and {Extended} {Eigenvectors}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {40--44},
     publisher = {mathdoc},
     volume = {90},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a4/}
}
TY  - JOUR
AU  - M. Gurdal
TI  - Connections between Deddens Algebras and Extended Eigenvectors
JO  - Matematičeskie zametki
PY  - 2011
SP  - 40
EP  - 44
VL  - 90
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a4/
LA  - ru
ID  - MZM_2011_90_1_a4
ER  - 
%0 Journal Article
%A M. Gurdal
%T Connections between Deddens Algebras and Extended Eigenvectors
%J Matematičeskie zametki
%D 2011
%P 40-44
%V 90
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a4/
%G ru
%F MZM_2011_90_1_a4
M. Gurdal. Connections between Deddens Algebras and Extended Eigenvectors. Matematičeskie zametki, Tome 90 (2011) no. 1, pp. 40-44. http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a4/

[1] A. Biswas, A. Lambert, S. Petrovic, “Extended eigenvalues and the Volterra operator”, Glasg. Math. J., 44:3 (2002), 521–534 | DOI | MR | Zbl

[2] M. T. Karaev, H. S. Mustafayev, “On some properties of Deddens algebras”, Rocky Mountain J. Math., 33:3 (2003), 915–926 | DOI | MR | Zbl

[3] A. Lambert, S. Petrovic, “Beyond hyperinvariance for compact operators”, J. Funct. Anal., 219:1 (2005), 93–108 | DOI | MR | Zbl

[4] S. Petrovic, “On the extended eigenvalues of some Volterra operators”, Integral Equations Operator Theory, 57:4 (2007), 593–598 | DOI | MR | Zbl