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


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.


Function spaces Weighted composition operators Topological vector spaces 

Mathematics Subject Classification

46E10 46E15 46E22 47B33 



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.


  1. 1.
    Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces, vol. 44. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  2. 2.
    Bartle, R.G.: On compactness in functional analysis. Trans. Am. Math. Soc. 79, 35–57 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourbaki, N.: Elements of Mathematics. General Topology. Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Massachusetts-London-Don Mills, Ontario (1966)Google Scholar
  4. 4.
    Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142(1), 289–299 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chirka, E.M.: Complex Analytic Sets. Mathematics and Its Applications (Soviet Series), vol. 46. Kluwer Academic Publishers Group, Dordrecht (1989). Translated from the Russian by R. A. M. HoksbergenCrossRefzbMATHGoogle Scholar
  6. 6.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)zbMATHGoogle Scholar
  7. 7.
    Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6, 2nd edn. Heldermann Verlag, Berlin (1989). Translated from the Polish by the authorzbMATHGoogle Scholar
  8. 8.
    Ferreira, J.C., Menegatto, V.A.: Positive definiteness, reproducing kernel Hilbert spaces and beyond. Ann. Funct. Anal. 4(1), 64–88 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fleming, R.J., Jamison, J.E.: Isometries on Banach Spaces: Function Spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129. Chapman & Hall/CRC, Boca Raton, FL (2003)zbMATHGoogle Scholar
  10. 10.
    Garrido, M.I., Jaramillo, J.A.: Variations on the Banach–Stone theorem. Extracta Math. 17 (2002), no. 3, 351–383. IV Course on Banach Spaces and Operators (Spanish) (Laredo, 2001)Google Scholar
  11. 11.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  12. 12.
    Jury, M.T., Martin, R.T.W.: Extremal multipliers of the Drury–Arveson space. arXiv:1608.04327 (2016)
  13. 13.
    Le, T.: Normal and isometric weighted composition operators on the Fock space. Bull. Lond. Math. Soc. 46(4), 847–856 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mashreghi, J., Ransford, T.: A Gleason–Kahane–Żelazko theorem for modules and applications to holomorphic function spaces. Bull. Lond. Math. Soc. 47(6), 1014–1020 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Narici, L., Beckenstein, E.: Topological Vector Spaces. Pure and Applied Mathematics, vol. 296, 2nd edn. CRC Press, Boca Raton, FL (2011)zbMATHGoogle Scholar
  16. 16.
    Singh, R.K., Manhas, J.S.: Composition Operators on Function Spaces. North-Holland Mathematics Studies, vol. 179. North-Holland Publishing Co., Amsterdam (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Singh, R.K., Summers, W.H.: Composition operators on weighted spaces of continuous functions. J. Aust. Math. Soc. Ser. A 45(3), 303–319 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wada, J.: Weakly compact linear operators on function spaces. Osaka Math. J. 13, 169–183 (1961)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zorboska, N.: Unitary and normal weighted composition operators on reproducing kernel Hilbert spaces of holomorphic functions. preprint (2017)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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