Superposition operators between weighted Banach spaces of analytic functions of controlled growth
For a weight \(\nu \), let \(H_{\nu}^{\infty}=H_{\nu}^{\infty}(\mathbb{D})\) denote the weighted Banach space of holomorphic functions on the unit disc \(\mathbb{D}\) on the complex plane equipped with the norm \(\| f\|_{\nu}:=\sup_{z\in\mathbb{D}}\nu(z) | f(z)|\). The paper is concerned with conditions on the entire function \(h\) which guarantee that the corresponding Nemytskij superposition operator \(S_{\varphi}(f) :=\varphi \circ f\) maps \(H_{\mu}^{\infty}\) into \(H_{\nu}^{\infty}\). The authors also show that all superposition operators generated by such entire functions are bounded and continuous. Moreover, they give some results for concrete weights (polynomial weights, exponential weights and logarithmic weights).
scientific article; zbMATH DE number 6176421
scientific article