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.
Similar content being viewed by others
References
Barroso, J.A.: Introduction to Holomorphy. North-Holland, New York (1985)
Beltrán, M.J.: Operators on weighted spaces of holomorphic functions. Thesis, Universitat Politècnica de València, València (2014)
Chacón, G.A., Chacón, G.R., Giménez, J.: Composition operators on spaces of entire functions. Proc. Am. Math. Soc. 135(7), 2205–2218 (2007)
Carando, D., Sevilla-Peris, P.: Spectra of weighted algebra of holomorphic function. Math. Z. 263, 887–902 (2009)
Carando, D.: A characterization of composition operators on algebras of analytic functions. Proc. Edinb. Math. Soc. 51, 305–313 (2008)
Carswell, B.J., MacCluer, B.D., Schuster, A.: Composition operators on the Fock space. Acta Sci. Math. (Szeged) 69(34), 871–887 (2003)
Chae, S.B.: Holomorphy and Calculas in Normed Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92. M. Dekker, New York (1985)
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)
Dineen, S.: Complex Analysis in Locally Convex Spaces. North-Holland, New York (1981)
Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)
Doan, M.L.: Hilbert spaces of entire functions and composition operators. Complex Anal. Oper. Theory 10(1), 213–230 (2016)
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)
Duyos-Ruiz, S.M.: On the existence of universal functions. Sov. Math. Dokl. 27, 9–13 (1983)
Galindo, P., Lindström, M., Ryan, R.: Weakly compact composition operators between algebras of bounded analytic functions. Proc. Am. Math. Soc. 128, 149–155 (2000)
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)
Gupta, M., Pradhan, S.: On Orlicz spaces of entire functions. Indian J. Pure Appl. Math. 39, 123–135 (2008)
Gupta, M., Pradhan, S.: Differential operators on the Orlicz spaces of entire functions. Ganita 61, 7–18 (2010)
Gupta, M., Baweja, D.: Weighted spaces of holomorphic functions on Banach spaces and the approximation property. Extr. Math. 31(2), 123–144 (2016)
Holub, J.R.: Reflexivity of \({\cal{L}}(E, F)\). Proc. Am. Math. Soc. 39, 175–177 (1973)
Kamthan, P.K., Gupta, M.: Sequence Spaces and Series. Lecture Notes No. 65, Marcel Dekker Inc., New York (1981)
Le, T.: Composition operators between Segal–Bargmann spaces. J. Oper. Theory 78(1), 135–158 (2017)
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)
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)
Mujica, J.: Complex Analysis in Banach Spaces. North-Holland, New York (1986)
Pietsch, A.: Nuclear Locally Convex Spaces. Springer, London (1972)
Pradhan, S.: Orlicz Spaces of Entire Functions and Modular Sequence Spaces. Ph.D. Thesis, IIT Kanpur, Kanpur, India (2008)
Ruckle, W.: Reflexivity of \({\cal{L}}(E, F)\). Proc. Am. Math. Soc. 34, 171–174 (1972)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
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)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-018-0229-7