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Collectanea Mathematica

, Volume 70, Issue 2, pp 323–346 | Cite as

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

  • Manjul Gupta
  • Deepika BawejaEmail author
Article
  • 67 Downloads

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.

Keywords

Holomorphic mappings Homogeneous polynomials Differentiation operators Translation operators 

Mathematics Subject Classification

46G20 46E50 

Notes

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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Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIIT KanpurKanpurIndia
  2. 2.Department of MathematicsBirla Institute of Technology and Science-Pilani, Hyderabad CampusHyderabadIndia

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