## 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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