The essential norm of weighted composition operators on weighted Banach spaces of analytic functions (Q421117)

The authors study the essential norm for weighted composition operators, defined by \(uC_{\phi}(f)=u (f\circ\phi)\) for given \(\phi,u\in H(\mathbb D)\) and \(\|\phi\|_\infty\leq 1\), acting on the spaces \(H^\infty_v\) with the norm \(\|f\|_v=\sup_{|z|<1}v(z)|f(z)|\) for a given continuous and bounded \(v:\mathbb D\to (0,\infty)\). The main result establishes that, for radial weights \(v,w\) with \[ \lim_{r\to 1} w(r)=\lim_{r\to 1} v(r)=0, \] one has that \[ \|uC_\phi\|_{e, H^\infty_v\to H^\infty_w}=\limsup_{n\to \infty} \frac{\|u\phi^n\|_w}{\|\xi^n\|_v}. \] Applications to composition operators acting on weighted Bloch spaces are given, generalizing those obtained by \textit{R.-H. Zhao} [``Essential norms of composition operators between Bloch type spaces'', Proc. Am. Math. Soc. 138, No. 7, 2537--2546 (2010; Zbl 1190.47028)] and also those by \textit{J. S. Manhas} and \textit{R.-H. Zhao} [``New estimates of essential norms of weighted composition operators between Bloch type spaces'', J. Math. Anal. Appl. 389, No. 1, 32--47 (2012; Zbl 1267.47042)].