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Essential norm of generalized composition operators on weighted Hardy spaces. (English) Zbl 1322.47032

For the open unit disk \(\mathbb{D}\) and the space of analytic functions \(H(\mathbb{D})\) on \(\mathbb{D}\), a positive continuous integrable function \(\omega\) on \([0,1)\) is called a weight if \(\omega(z)=\omega(|z|)\) for every \(z \in \mathbb{D}\). A weight \(\omega\) is called an almost standard if it is non-decreasing and \(\frac{\omega(r)}{(1-r)^{1+\gamma}}\) is non-decreasing for some \(\gamma>0\). The weighted Hardy space \(H_{\omega}\) is defined as \[ H_{\omega} =\left\{ f\in H(\mathbb{D}) : ||f||_{H_{\omega}}^2=|f(0)|^2+\int_{\mathbb{D}}|f'(z)|^2\omega(z)\,dA(z)< \infty \right\}, \] where \(dA\) is the normalized area measure on \(\mathbb{D}\). For \(g \in H(\mathbb{D})\) and a holomorphic self-map \(\varphi\) on \(\mathbb{D}\), the generalized composition operator \(J_{g, \varphi}\) on \(H(\mathbb{D})\) is given by \[ J_{g, \varphi} = \int_0^z f'(\varphi(\varsigma))g(\varsigma)\,d\varsigma. \] \(J_{g, \varphi}\) extends both the integral-type (Volterra-type) operator \(J_g\) (as \(J_g = J_{g,\operatorname{id}_{\mathbb{D}}}\) ) and composition operators (\( J_{\varphi', \varphi} = C_{\varphi} - \delta_{\varphi(0)}\), where \(\delta_{\varphi(0)}\) is the point-evaluation functional at \(\varphi(0)\)). The author applies the Careleson-type criterion on \(H_{\omega}\), as established in [S. Stević and the author, Abstr. Appl. Anal. 2011, Article ID 698038, 16 p. (2011; Zbl 1236.47026)], to prove boundedness and to estimate the upper and lower bounds for the essential norms of \(J_{g, \varphi}\), which leads to a characterization of the compactness of \(J_{g, \varphi}\).

MSC:

47B33 Linear composition operators
46E10 Topological linear spaces of continuous, differentiable or analytic functions
30H10 Hardy spaces

Citations:

Zbl 1236.47026
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