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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1441–1464 | Cite as

Continuity and Holomorphicity of Symbols of Weighted Composition Operators

  • Eugene BilokopytovEmail author
Article
  • 56 Downloads

Abstract

The main problem considered in this article is the following: if \({\mathbf {F}}\), \({\mathbf {E}}\) are normed spaces of continuous functions over topological spaces X and Y respectively, and \(\omega :Y\rightarrow {\mathbb {C}}\) and \(\Phi :Y\rightarrow X\) are such that the weighted composition operator \(W_{\Phi ,\omega }\) is continuous from \({\mathbf {F}}\) into \({\mathbf {E}}\), when can we guarantee that both \(\Phi \) and \(\omega \) are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven before for more concrete function spaces.

Keywords

Function spaces Weighted composition operators Topological vector spaces 

Mathematics Subject Classification

46E10 46E15 46E22 47B33 

Notes

Acknowledgements

This paper is a part of the author’s thesis and he wants to thank his supervisor Nina Zorboska for the general guidance and some valuable insights regarding Sect. 4. Also the author wants to thank Daniel Fischer, who contributed to the proof of Proposition 5.1 and the service Math.Stackexchange which made it possible.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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