A new essential norm estimate of composition operators from weighted Bloch space into \(\mu\)-Bloch spaces (Q2443762)

Summary: Let \(\mu\) be any weight function defined on the unit disk \(\mathbb D\) and let \(\phi\) be an analytic self-map of \(\mathbb D\). In the present paper, we show that the essential norm of composition operator \(C_\phi\) mapping from the weighted Bloch space to \(\mu\)-Bloch space \(\mathcal B^\mu\) is comparable to \(\lim \sup_{|\alpha|\to 1^{-}}||\sigma_a \circ \phi||_{\mathcal B^{\mu}}\) where for \(a \in \mathbb D\), \(\sigma_a\) is a certain special function in the weighted Bloch space. As a consequence of our estimate, we extend the results about the compactness of composition operators due to \textit{M. Tjani} [Trans. Am. Math. Soc. 355, No. 11, 4683--4698 (2003; Zbl 1045.47020)].