Composition and translation operators on certain subspaces of the space of entire functions of bounded type
- Autores: Manjul Gupta, Deepika Baweja
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 2, 2019, págs. 323-346
- Idioma: inglés
- DOI: 10.1007/s13348-018-0229-7
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Resumen
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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