## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Advance publication (2019), 22 pages.

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

Shuming Wang, Maofa Wang, and Xin Guo

#### 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.

#### Article information

**Source**

Taiwanese J. Math., Advance publication (2019), 22 pages.

**Dates**

First available in Project Euclid: 19 April 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1555639218

**Digital Object Identifier**

doi:10.11650/tjm/190405

**Subjects**

Primary: 47B33: Composition operators

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 47B65: Positive operators and order-bounded operators

**Keywords**

product-type operator weighted differentiation composition operator weighted-type space order boundedness essential norm

#### Citation

Wang, Shuming; Wang, Maofa; Guo, Xin. Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions. Taiwanese J. Math., advance publication, 19 April 2019. doi:10.11650/tjm/190405. https://projecteuclid.org/euclid.twjm/1555639218