Resolvent estimates and decomposable extensions of generalized Cesàro operators
Let \(H(\mathbb D)\) be the space of all holomorphic functions in the unit disk \(\mathbb D\). For \(g\in H(\mathbb D)\), the generalized Cesàro operator \({\mathfrak C}_g\) is defined by \[ {\mathfrak C}_g f(z)=\frac{1}{z}\int_0^z f(w)g'(w) \,dw, \quad z\in\mathbb D,\;f\in H(\mathbb D). \] The authors determine the spectrum of generalized Cesàro operators for special symbols on various spaces of analytic functions on \(\mathbb D\), including the Hardy spaces \(H^p\), the standard weighted Bergman spaces \[ A^{p,\alpha}= \bigg\{f\in H(\mathbb D): \int_{\mathbb D} |f(z)|^p(1-|z|)^\alpha \,d\sigma_2(z)<\infty\bigg\}, \] the weighted Dirichlet spaces \[ D^{p,\alpha}=\{f\in H(\mathbb D): f '\in A^{p,\alpha}\}, \] as well as the growth classes \[ A^{-\gamma}= \bigg\{f\in H(\mathbb D): \sup_{z\in\mathbb D}(1-|z|)^{\gamma}|f(z)|<\infty\bigg\} \] and its closed subspace \[ A^{-\gamma_0}= \bigg\{f\in A^{-\gamma}: \limsup_{|z|\to 1}(1-|z|)^{\gamma}|f(z)|=0\bigg\}. \] Results on boundedness and compactness of these operators as well as certain weighted composition operators are included, too. Conditions are given under which certain types of these Cesàro operators are subdecomposable. The paper contains a great variety of other interesting material.
scientific article; zbMATH DE number 5655670
scientific article