On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra. (English) Zbl 1222.47041

A complex number \(\lambda\) is an extended eigenvalue of a bounded operator \(X\) on a Banach space if there is a nonzero bounded operator \(Y\) such that \(XY = \lambda YX\). In this case, \(Y\) is said to be an extended eigenvector associated with the eigenvalue \(\lambda\). A function \( f(z) = \sum_{n=0}^{\infty} a_n z^n\) is in the Wiener algebra \(W(\mathbb D)\) if it is analytic on the unit disk \(\mathbb D\) and \( f(z) = \sum_{n=0}^{\infty} | a_n | < \infty\).
The author’s abstract and main result claim that the set of extended eigenvalues of the shift operator \(S=M_z\) on \(W(\mathbb D)\) is equal to the closed unit disk \(\bar{\mathbb D }\). This is actually not the case. In fact, the author correctly observes prior to the main result that \(\lambda =0\) cannot be an extended eigenvalue of \(S\). The author’s result is essentially the observation that if \(\lambda \in \bar{\mathbb D} \setminus \{0\}\), \(g\) is any nonzero function in the multiplier algebra of \(W(\mathbb D)\), and \(C_\lambda\) denotes the composition operator \(C_\lambda(f) = f(\lambda z)\), then \(A=M_gC_\lambda\) is an extended eigenvector for \(S\) associated with \(\lambda\).


47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Full Text: DOI


[1] Biswas, A.; Lambert, A.; Petrovic, S., Extended eigenvalues and the Volterra operator, Glasg. math. J., 44, 521-534, (2002) · Zbl 1037.47013
[2] Biswas, A.; Petrovic, S., On extended eigenvalues of operators, Integral equations and operator theory, 55, 233-248, (2006) · Zbl 1119.47019
[3] Lambert, A., Hyperinvariant subspaces and extended eigenvalues, New York J. math., 10, 83-88, (2004) · Zbl 1090.47003
[4] Karaev, M.T.; Mustafayev, H.S., On some properties of deddens algebras, Rocky mountain J. math., 33, 915-926, (2003) · Zbl 1079.46032
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.