Abstract
Let \({\text {Lip}}^n(X, \alpha )\) be the algebra of complex-valued functions on a perfect compact plane set X, whose derivatives up to order n exist and satisfy the Lipschitz condition of order \(0<\alpha \le 1\). We establish a necessary and sufficient condition for a weighted composition operator on \({\text {Lip}}^n(X, \alpha )\) to be compact. To obtain the necessary condition in the case \(0<\alpha < 1\), we provide a relation between these algebras and Zygmund-type spaces \(\mathcal {Z}_n^\alpha \). We then conclude some interesting results about weighted composition operators on \(\mathcal {Z}_n^\alpha \) and determine the spectra of these operators when they are compact or Riesz.
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Communicated by Hamid Reza Ebrahimi Vishki.
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Amiri, S., Golbaharan, A. & Mahyar, H. Weighted Composition Operators on Differentiable Lipschitz Algebras. Bull. Iran. Math. Soc. 44, 955–968 (2018). https://doi.org/10.1007/s41980-018-0062-5
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DOI: https://doi.org/10.1007/s41980-018-0062-5
Keywords
- Differentiable Lipschitz functions
- Weighted composition operators
- Bloch- and Zygmund-type spaces
- Compact and Riesz operators
- Spectra