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Composition and translation operators on certain subspaces of the space of entire functions of bounded type

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Abstract

In this paper, for complex Banach spaces E, F and \(1\le p\le \infty \), the subspaces \({\mathcal {H}}_{p}^{\gamma }(E,F)\) of the space \({\mathcal {H}}_{b}(E,F)\) consisting of holomorphic mappings of bounded type from E into F, have been introduced and studied. Here the notation \(\gamma \) stands for a comparison function \(\gamma \) which is an entire function defined on the complex plane, as \(\gamma (z)=\sum \nolimits _{n=0}^{\infty } \gamma _{n} z^{n}, \gamma _{n} >0\) for each \(n \in {\mathbb {N}}_{0}\) with \(\gamma _{n}^{\frac{1}{n}}\rightarrow 0\) and \(\frac{\gamma _{n+1}}{\gamma _{n}} \downarrow 0\) as n increases to \(\infty \). Besides considering the relationships amongst these spaces, their vector valued sequential analogues have also been obtained for \(1\le p <\infty \). These results are used in obtaining the dual and Schauder decomposition of \({\mathcal {H}}_{p}^{\gamma }(E,F)\), \(1\le p <\infty \). The continuity of differentiation and translation operator has been proved by restricting \(\gamma \) suitably and the spectrum of the differentiation operator \(D_a\) has been investigated. Finally, the continuity and compactness of the composition operator \(C_{\phi }\), defined corresponding to a holomorphic function \(\phi \) have been investigated.

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References

  1. Barroso, J.A.: Introduction to Holomorphy. North-Holland, New York (1985)

    MATH  Google Scholar 

  2. Beltrán, M.J.: Operators on weighted spaces of holomorphic functions. Thesis, Universitat Politècnica de València, València (2014)

  3. Chacón, G.A., Chacón, G.R., Giménez, J.: Composition operators on spaces of entire functions. Proc. Am. Math. Soc. 135(7), 2205–2218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carando, D., Sevilla-Peris, P.: Spectra of weighted algebra of holomorphic function. Math. Z. 263, 887–902 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carando, D.: A characterization of composition operators on algebras of analytic functions. Proc. Edinb. Math. Soc. 51, 305–313 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carswell, B.J., MacCluer, B.D., Schuster, A.: Composition operators on the Fock space. Acta Sci. Math. (Szeged) 69(34), 871–887 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Chae, S.B.: Holomorphy and Calculas in Normed Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92. M. Dekker, New York (1985)

    Google Scholar 

  8. Chan, K.C., Shapiro, J.H.: The cyclic behaviour of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40, 1421–1449 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dineen, S.: Complex Analysis in Locally Convex Spaces. North-Holland, New York (1981)

    MATH  Google Scholar 

  10. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)

    Book  MATH  Google Scholar 

  11. Doan, M.L.: Hilbert spaces of entire functions and composition operators. Complex Anal. Oper. Theory 10(1), 213–230 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Doan, M.L., Khoi, L.H.: Composition operators on Hilbert spaces of entire functions. C. R. Math. Acad. Sci. Paris Ser. I 353(6), 495–499 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duyos-Ruiz, S.M.: On the existence of universal functions. Sov. Math. Dokl. 27, 9–13 (1983)

    MathSciNet  MATH  Google Scholar 

  14. Galindo, P., Lindström, M., Ryan, R.: Weakly compact composition operators between algebras of bounded analytic functions. Proc. Am. Math. Soc. 128, 149–155 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. García, D., Maestre, M., Sevilla-Peris, P.: Composition operators between weighted spaces of holomophic functions on Banach spaces. Ann. Acad. Sci. Fenn. Math. 29, 81–98 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Gupta, M., Pradhan, S.: On Orlicz spaces of entire functions. Indian J. Pure Appl. Math. 39, 123–135 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gupta, M., Pradhan, S.: Differential operators on the Orlicz spaces of entire functions. Ganita 61, 7–18 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Gupta, M., Baweja, D.: Weighted spaces of holomorphic functions on Banach spaces and the approximation property. Extr. Math. 31(2), 123–144 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Holub, J.R.: Reflexivity of \({\cal{L}}(E, F)\). Proc. Am. Math. Soc. 39, 175–177 (1973)

    MATH  Google Scholar 

  20. Kamthan, P.K., Gupta, M.: Sequence Spaces and Series. Lecture Notes No. 65, Marcel Dekker Inc., New York (1981)

  21. Le, T.: Composition operators between Segal–Bargmann spaces. J. Oper. Theory 78(1), 135–158 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Matos, M.C.: On the Fourier-Borel transformation and spaces of entire functions in a normed space. In: Zapata, G.I. (ed.) Functional Analysis, Holomorphy and Approximation Theory II, pp. 139–170. North-Holland, New York (1984)

    Chapter  Google Scholar 

  23. Matos, M.C.: On convolution operators in spaces of entire functions of given type and order. In: Mujica, J. (ed.) Complex Analysis, Functional Analysis and Approximation Theory, pp. 129–171. North-Holland, New York (1986)

    Google Scholar 

  24. Mujica, J.: Complex Analysis in Banach Spaces. North-Holland, New York (1986)

    MATH  Google Scholar 

  25. Pietsch, A.: Nuclear Locally Convex Spaces. Springer, London (1972)

    Book  MATH  Google Scholar 

  26. Pradhan, S.: Orlicz Spaces of Entire Functions and Modular Sequence Spaces. Ph.D. Thesis, IIT Kanpur, Kanpur, India (2008)

  27. Ruckle, W.: Reflexivity of \({\cal{L}}(E, F)\). Proc. Am. Math. Soc. 34, 171–174 (1972)

    MATH  Google Scholar 

  28. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  29. Stochel, J., Stochel, J.B.: Composition operators on Hilbert spaces of entire functions with analytic symbols. J. Math. Anal. Appl. 454(2), 1019–1066 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank referee for suggesting modifications in Theorem 5.2, Theorem 5.8 and Proposition 6.1(a) besides bringing to the notice of authors the references [6, 7, 21, 29]. The second author acknowledges the Council of Scientific and Industrial Research INDIA for a Research Fellowship (Grant No. 09/092(0843)/2012-EMR-I).

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Correspondence to Deepika Baweja.

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Gupta, M., Baweja, D. Composition and translation operators on certain subspaces of the space of entire functions of bounded type. Collect. Math. 70, 323–346 (2019). https://doi.org/10.1007/s13348-018-0229-7

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