On k-composition and k-Hankel composition operators on the derivative Hardy space

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.