Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Abstract

Let $H(\mathbb{D})$ denote the space of holomorphic functions on the unit disk $\mathbb{D}$ of $\mathbb{C}$, $\psi,\varphi \in H(\mathbb{D})$, $\varphi(\mathbb{D}) \subset \mathbb{D}$ and $n \in \mathbb{N} \cup \{0\}$. Let $C_{\varphi}$, $M_{\psi}$ and $D^n$ denote the composition, multiplication and iterated differentiation operators, respectively. To treat the operators induced by products of these operators in a unified manner, we introduce a sum operator $\sum_{j=0}^n M_{\psi_j} C_{\varphi} D^j$. We characterize the boundedness and compactness of this sum operator mapping from a large class of Banach spaces of holomorphic functions into the $k$th weighted-type space $\mathcal{W}_{\mu}^{(k)}$ (or $\mathcal{W}_{\mu,0}^{(k)}$), $k \in \mathbb{N} \cup \{0\}$, and give its estimates of norm and essential norm. Our results show that the boundedness and compactness of the sum operator depend only on the symbols and the norm of the point-evaluation functionals on the domain space. Our results cover many known results in the literature. Moreover, we introduce the order boundedness of the sum operator and turn its study into that of the boundedness and compactness.