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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 935–958 | Cite as

Integral, Differential and Multiplication Operators on Generalized Fock Spaces

  • Tesfa MengestieEmail author
  • Sei-Ichiro Ueki
Article

Abstract

Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane \(\mathbb {C}\). The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols g which induce bounded Volterra companion integral \(I_g\) and multiplication operators \(M_g\) acting between the weighted spaces. We also describe the bounded and compact Volterra-type integral operators \(V_g\) acting between \({\mathcal {F}}_q^\psi \) and \({\mathcal {F}}_p^\psi \) when at least one of the exponents p or q is infinite, and extend results of Constantin and Peláez for finite exponent cases. Furthermore, we showed that the differential operator D acts in unbounded fashion on these and the classical Fock spaces.

Keywords

Weighted Fock space Generalized Fock spaces Volterra operator Multiplication operator Differential operator Bounded Compact 

Mathematics Subject Classification

Primary 47B32 30H20 Secondary 46E22 46E20 47B33 

Notes

Acknowledgements

We would like to thank the referee for careful review of our paper and pointing us relevant literatures, which eventually helped us put our work in context to already known results.

References

  1. 1.
    Abakumov, E., Doubtsov, E.: Volterra type operators on growth Fock spaces. Arch. Math. (Basel) 108(4), 383–393 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290, 1144–1162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aleman, A.: A Class of Integral Operators on Spaces of Analytic Functions. Topics in Complex Analysis and Operator Theory, pp. 3–30. University of Málaga, Málaga (2007)zbMATHGoogle Scholar
  4. 4.
    Aleman, A., Cima, J.: An integral operator on \(H^p\) and Hardy’s inequality. J. Anal. Math. 85, 157–176 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aleman, A., Siskakis, A.: An integral operator on \(H^p\). Complex Var. 28, 149–158 (1995)zbMATHGoogle Scholar
  6. 6.
    Aleman, A., Siskakis, A.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46, 337–356 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288, 1216–1225 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borichev, A., Dhuez, R., Kellay, K.: Sampling and interpolation in large Bergman and Focks paces. J. Funct. Anal. 242, 563–606 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Constantin, O.: Volterra type integration operators on Fock spaces. Proc. Am. Math. Soc. 140(12), 4247–4257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Constantin, O., Peláez, José Ángel: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted fock spaces. J. Geom. Anal. 26, 1109–1154 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Rev. Mat. Iberoam. 3, 61–138 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Luecking, D.: Embedding theorems for space of analytic functions via Khinchine’s inequality. Mich. Math. J. 40, 333–358 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Marco, N., Massaneda, M., Ortega-Cerdà, J.: Interpolating and sampling sequences for entire functions. Geom. Funct. Anal. 13, 862–914 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mengestie, T.: Generalized Volterra companion operators on Fock spaces. Potential Anal. 44, 579–599 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mengestie, T.: Volterra type and weighted composition operators on weighted Fock spaces. Integral Equ. Oper. Theory 76(1), 81–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oleinik, V.L.: Embedding theorems for weighted classes of harmonic and analytic functions. J. Math. Sci. 9(2), 228–243 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pau, J., Peláez, J.A.: Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weightes. J. Funct. Anal. 259(10), 2727–2756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pommerenke, C.: Schlichte Funktionen und analytische Funktionen von beschránkter mittlerer Oszillation. Comment. Math. Helv. 52(4), 591–602 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rashkovskii, A.: Classical and new loglog-theorems. Expo. Math. 27(4), 271–287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Seip, K., Youssfi, E.: Hankel operators on Fock spaces and related Bergman kernel estimates. J. Geom. Anal. 23, 170–201 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Siskakis, A.: Volterra operators on spaces of analytic functions-a survey. In: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, pp. 51–68. University of Seville, Seville (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesWestern Norway University of Applied SciencesStordNorway
  2. 2.Toki UniversityHitachiJapan

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