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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 \(\lambda\).

MSC:

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.)
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References:

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