Results in Mathematics

, 74:23 | Cite as

A Left Linear Weighted Composition Operator on Quaternionic Fock Space

  • Yu-Xia LiangEmail author


A left linear weighted composition operator \(W_{f,\varphi }\) is defined on slice regular quaternionic Fock space \(\mathcal {F}^2(\mathbb {H})\). We carry out a comprehensive analysis on its classical properties. Firstly, the boundedness and compactness of weighted composition operator on \(\mathcal {F}^2(\mathbb {H})\) are investigated systematically, which can be seen new and brief characterizations. And then all normal bounded weighted composition operators are found, particularly, equivalent conditions for self-adjoint weighted operators on \(\mathcal {F}^2(\mathbb {H})\) are developed. Finally, we describe all types of isometric weighted composition operators on \(\mathcal {F}^2(\mathbb {H})\).


Weighted composition operator quaternionic Fock space boundedness compactness self-adjoint isometry 

Mathematics Subject Classification

Primary 30G35 47B38 



Y. X. Liang is supported by the National Natural Science Foundation of China (Grant No. 11701422).


  1. 1.
    Adler, S.: Quaternionic Quantum Field Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  2. 2.
    Alpay, D., Colombo, F., Sabadini, I., Salomon, G.: The Fock Space in the Slice Hyperholomorphic Setting, pp. 43–59. Springer, Berlin (2014)zbMATHGoogle Scholar
  3. 3.
    Colombo, F., Gonzalez-Cervantes, J.O., et al.: On two approaches to the Bergman theory for slice regular functions. In: Advances in Hypercomplex Analysis, vol. 1, pp. 39–54. Springer INdAM Series, Berlin (2013)Google Scholar
  4. 4.
    Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics, vol. 39. Birkhäuser, Boston (2004)CrossRefGoogle Scholar
  5. 5.
    Colombo, F., Sabadini, I., Struppa, D.C.: Slice Monogenic Functions, Noncommutative Functional Calculus. Springer, Basel (2011)CrossRefGoogle Scholar
  6. 6.
    Colombo, F., Sabadini, I., Struppa, D.C.: Entire Slice Regular Functions. Springer, Berlin (2016)CrossRefGoogle Scholar
  7. 7.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  8. 8.
    Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and \(n\)-Dimensional Space. Birkhäuser, Basel (2008)zbMATHGoogle Scholar
  9. 9.
    Horwitz, L.P., Razon, A.: Tensor product of quaternion Hilbert modules. In: Classical and Quantum Systems (Goslar), pp. 266–268 (1991)Google Scholar
  10. 10.
    Kumar, S., Manzoor, K.: Slice regular Besov spaces of hyperholomorphic functions and composition operators (2016). arXiv:1609.02394v1
  11. 11.
    Kumar, S., Manzoor, K., Singh, P.: Composition operators in Bloch spaces of slice hyperholomorphic functions. Adv. Appl. Clifford Algebra 27(2), 1459–1477 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lian, P., Liang, Y.X.: Weighted composition operator on quaternionic Fock space, arXiv: 1803.0678v1, pp. 1–39. (2018)
  13. 13.
    Le, T.: Normal and isometric weighted composition operators on the Fock space. Bull. Lond. Math. Soc. 46(4), 847–856 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sabadini, I., Saracco, A.: Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball. J. Lond. Math. Soc. 95(3), 853–874 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)CrossRefGoogle Scholar
  16. 16.
    Villalba, C., Colombo, F., Gantner, J., et al.: Bloch, Besov and Dirichlet spaces of slice hyperholomorphic functions. Complex Anal. Oper. Theory 9(2), 479–517 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhao, L., Pang, C.: A class of weighted composition operators on the Fock space. J. Math. Res. Appl. 35(3), 303–310 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhu, K.H.: Analysis on Fock Spaces. Springer, New York (2012)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTianjin Normal UniversityTianjinPeople’s Republic of China

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