Isometric weighted composition operators on weighted Banach spaces of type \(H^{\infty }\). (English) Zbl 1154.47017
The authors study conditions for the weighted composition operator \(C_{\varphi, \psi}(f)=\psi (f\circ \varphi)\) to be an isometry on the weighted space \(H^\infty_v\) defined by analytic functions in the disk such that \(\sup_{|z|<1}v(z)|f(z)|<\infty\).
Their results extends those by [M.J.Martin and D.Vukotić, Bull.Lond.Math.Soc.39, No.1, 151–155 (2007; Zbl 1115.47024); Contemporary Mathematics 393, 133–138 (2006; Zbl 1121.47018)] for composition operators on Bloch spaces. The main results are stated for radial and decreasing weights and establish sufficient conditions for \(C_{\varphi, \psi}\) to be an isometry on \(H^\infty_v\) under the extra assumptions that \(v\) is continuously differentiable with respect to \(|z|\) and satisfies \(\inf_k \frac{v(1- 2^{-(k+1)})}{v(1- 2^{-k})}>0\), and necessary conditions for weights such that \(w(z)=\frac{v(z)}{(1-|z|^2)^p}\) is a weight which coincides with its associate \(\tilde w\) for some \(0<p<\infty\). Some additional results for the case \(\psi=1\) are given under weaker assumptions on the weight.
Their results extends those by [M.J.Martin and D.Vukotić, Bull.Lond.Math.Soc.39, No.1, 151–155 (2007; Zbl 1115.47024); Contemporary Mathematics 393, 133–138 (2006; Zbl 1121.47018)] for composition operators on Bloch spaces. The main results are stated for radial and decreasing weights and establish sufficient conditions for \(C_{\varphi, \psi}\) to be an isometry on \(H^\infty_v\) under the extra assumptions that \(v\) is continuously differentiable with respect to \(|z|\) and satisfies \(\inf_k \frac{v(1- 2^{-(k+1)})}{v(1- 2^{-k})}>0\), and necessary conditions for weights such that \(w(z)=\frac{v(z)}{(1-|z|^2)^p}\) is a weight which coincides with its associate \(\tilde w\) for some \(0<p<\infty\). Some additional results for the case \(\psi=1\) are given under weaker assumptions on the weight.
Reviewer: Oscar Blasco (Valencia)
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