Abstract
In this paper, we propose the generalized derivative Hardy space $$S^2_{\alpha ,\beta }(\mathbb {D})$$ which consists of functions whose derivatives are in the Hardy and Bergman spaces. In particular, we state basic results for $$S^2_{\alpha ,\beta }(\mathbb {D})$$ and focus on m-isometric multiplication operators. Moreover, we consider the complete Pick property in $$S_{\alpha ,\beta }^2(\mathbb {D})$$ and several applications of having the complete Pick property, which is related to the multiplication operators and composition operators. Finally, we study the Toeplitz operators on $$S^2_{\alpha ,\beta }(\mathbb {D})$$ and investigate a necessary and sufficient condition for the hyponormality of Toeplitz operator $$T_{\varphi }$$ on $$S^2_{\alpha ,\beta }(\mathbb {D})$$ .
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