Communications of the
Korean Mathematical Society CKMS

Let $\mathcal{H}(\mathbb{D})$ be the space of analytic functions on the unit disc $\mathbb{D}$. Let $\psi=(\psi_j)_{j=0}^n$ and $\Phi=(\Phi_j)_{j=0}^n$ be such that $\psi_j, \Phi_j \in \mathcal{H}(\mathbb{D})$. The linear differential operator is defined by $ T_{\psi}(f)=\sum_{j=0}^n \psi_j f^{(j)},$ $f\in \mathcal{H}(\mathbb{D})$. We characterize the boundedness and compactness of the difference operator $ (T_{\psi}-T_{\Phi})(f)=\sum_{j=0}^n \left(\psi_j-\Phi_j\right) f^{(j)}$ between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).

Keywords: Difference operators, differential operators, multiplication operators, weighted-type spaces, Bloch-type spaces, bounded and compact operators

MSC numbers: Primary 47B38, 47B33