Abstract
Let \(\theta : {\mathbb {N}}_0 \rightarrow {\mathbb {N}}_0\) be a function and \(k \in {\mathbb {N}}_0 \cup \{\infty \}\), the k-composition operator is a linear operator \(C_\theta ^k\) defined on derivative Hardy space \({\mathcal {S}}^2({\mathbb {D}})\) by \(C_\theta ^k (f) = \sum _{n=0}^k f_{\theta (n)}z^n\) for \(f(z) = \sum _{n=0}^\infty f_n z^n \text { in } {\mathcal {S}}^2({\mathbb {D}})\). Some basic properties of k-composition operators are studied. The k-composition operators have been extended to define k-Hankel composition operators on \({\mathcal {S}}^2({\mathbb {D}})\). The necessary and sufficient conditions are obtained for k-Hankel composition operators to be bounded or compact. The conditions for which k-Hankel composition operators commute are also explored. In addition to this, the necessary and sufficient condition for k-Hankel composition operators to be Hilbert–Schmidt is investigated.
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The authors are grateful to the referees for their valuable suggestions and comments which help us in improving the manuscript. Support of CSIR-UGC Research Grant(UGC) [Ref. No.: 21/12/2014(ii) EU-V, Sr. No. 2121440601] to the second author for carrying out the research work is gratefully acknowledged.
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Communicated by Deguang Han.
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Gupta, A., Gupta, B. On k-composition and k-Hankel composition operators on the derivative Hardy space. Banach J. Math. Anal. 14, 1602–1629 (2020). https://doi.org/10.1007/s43037-020-00080-z
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DOI: https://doi.org/10.1007/s43037-020-00080-z
Keywords
- k-Composition operator
- Derivative Hardy space
- Hankel operator
- k-Hankel composition operator
- Hilbert–Schmidt operator