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.

中文翻译：

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关于导数Hardy空间上的k-组合和k-Hankel组合算子

令 $$\theta : {\mathbb {N}}_0 \rightarrow {\mathbb {N}}_0$$ 是一个函数并且 $$k \in {\mathbb {N}}_0 \cup \{\infty \ }$$，k 组合算子是一个线性算子 $$C_\theta ^k$$ 定义在导数哈代空间 $${\mathcal {S}}^2({\mathbb {D}})$$ $$C_\theta ^k (f) = \sum _{n=0}^k f_{\theta (n)}z^n$$ for $$f(z) = \sum _{n=0} ^\infty f_n z^n \text { in } {\mathcal {S}}^2({\mathbb {D}})$$。研究了 k 复合算子的一些基本性质。k-组合算子已经扩展为在 $${\mathcal {S}}^2({\mathbb {D}})$$ 上定义 k-Hankel 组合算子。得到k-Hankel复合算子有界或紧致的充要条件。还探讨了 k-Hankel 组合算子通勤的条件。除此之外，