Essential norm of weighted composition operators from \(H^{\infty}\) to the Zygmund space (Q2828085)

For the open unit disk \(\mathbb{D} \subset \mathbb{C}\), the Bloch space \(\mathcal{B}\) is the space of all analytic functions \(f \in H(\mathbb{D})\) with norm NEWLINE\[NEWLINE||f||_{\mathcal{B}} = |f(0)| +\sup\limits_{z \in \mathbb{D}}(1-|z|^2)|f'(z)| < \inftyNEWLINE\]NEWLINE Let \(\mathcal{Z}\) denote the set of all functions \(f \in H(\mathbb{D}) \cap C(\overline{\mathbb{D}})\) such that NEWLINE\[NEWLINE\sup\limits_{\theta \in \mathbb{R},h > 0 } \frac{|f(e^{i(\theta+h)})+f(e^{i(\theta-h)})-2f(e^{i\theta})|}{h} < \inftyNEWLINE\]NEWLINE It follows from a theorem of Zygmund [\textit{P. L. Duren}, Theory of \(H^ p\) spaces. New York and London: Academic Press XII (1970; Zbl 0215.20203)] and the Closed Graph Theorem that \(f \in \mathcal{Z}\) if and only if \(\sup_{z \in \mathbb{D}}(1-|z|^2)|f''(z)| < \infty\). The set \(\mathcal{Z}\), called the Zygmund space, forms a Banach space under the norm given by NEWLINE\[NEWLINE||f||_{\mathcal{Z}} = |f(0)| + |f'(0)|+\sup\limits_{z \in \mathbb{D}}(1-|z|^2)|f''(z)|NEWLINE\]NEWLINE For Banach spaces \(X\) and \(Y\) and the essential norm of a bounded linear operator \(T : X \rightarrow Y\) is given by NEWLINE\[NEWLINE||T||_{e,X \rightarrow Y} = \inf\{||T-K||_{X \rightarrow Y} : K \in {\mathcal{K}} \}NEWLINE\]NEWLINE where \(\mathcal{K}\) is the set of all compact operators \(K : X \rightarrow Y\). It is well known that \(||T||_{e,X \rightarrow Y} = 0\) if and only if \(T : X \rightarrow Y\) is compact. Let \(H^{\infty} = H^{\infty}(\mathbb{D})\) be the space of all bounded analytic functions \(f\) on \(\mathbb{D}\) with norm \(||f||_{\infty}=\sup_{z \in \mathbb{D}}|f(z)|\). Composition operators and weighted composition operators on the Zygmund space were studied by several authors. Compactness of weighted composition operators \(uC_{\varphi} : H^{\infty} \rightarrow \mathcal{Z}\) were characterized in [\textit{F. Colonna} and \textit{S. Li}, Complex Anal. Oper. Theory 7, No. 5, 1495--1512 (2013; Zbl 1275.47050)]. In particular, it was shown that \(uC_{\varphi} : H^{\infty} \rightarrow \mathcal{Z}\) is compact if and only if \(\lim\limits_{n \rightarrow \infty} ||u\varphi^n||_{\mathcal{Z}} = 0\). Motivated by these results, the authors give a complete characterization of the essential norm of the operator \(uC_{\varphi} : H^{\infty} \rightarrow \mathcal{Z}\). Indeed, for a bounded weighted composition operator \(uC_{\varphi} : H^{\infty} \rightarrow \mathcal{Z}\), they prove that \(||uC_{\varphi}||_{e, H^{\infty}} \approx \lim\limits_{n \rightarrow \infty}\sup ||u\varphi^n||_{\mathcal{Z}}\).