## The essential norm of weighted composition operators on weighted Banach spaces of analytic functions.(English)Zbl 1252.47026

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 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 J. S. Manhas and 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)].

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general)

### Keywords:

essential norm; weighted composition operator

### Citations:

Zbl 1190.47028; Zbl 1267.47042
Full Text:

### References:

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