Duality of Bloch spaces and norm convergence of Taylor series
Let B be the Bloch space of the unit disc \({\mathbb{D}}\) with the norm \[ | f| =| f(0)| +\sup \{(1-| z|^ 2)| f'(z)|,\quad z\in {\mathbb{D}}\}. \] The Little Bloch space \(B_ 0\) consists of all holomorphic f(z) in \({\mathbb{D}}\) such that \((1-| z|^ 2)f(z)\) belongs to \(C_ 0({\mathbb{D}})\), the space of complex continuous functions on \({\bar {\mathbb{D}}}\) which vanish on \(\partial {\mathbb{D}}.\) For \(\alpha >-1\) the weighted Bergman space \(L_ a'(dA_{\alpha})\) is the subspace of \(L'({\mathbb{D}},dA_{\alpha})\) of all analytic functions, where \(dA_{\alpha}(z)=(1+\alpha)(1-| z|^ 2)^{\alpha}dA(z)\) and dA(z) is the area measure on \({\mathbb{D}}\). The author proves that under the \(pairing\) \(<f,g>_{\alpha}=\int_{{\mathbb{D}}}f(z)\overline{g(z)}dA(z)\) the following dualities hold (with equivalent norms) \[ B^*_ 0\cong L_ a'(dA_{\alpha})\text{ and } (L_ a'(dA_{\alpha}))^*\cong B. \] This is a generalization of the same equalities known for \(\alpha =0\). Moreover it is shown that there exist functions \(f\in B_ 0\) and \(g\in L_ a'(dA_{\alpha})\), whose Taylor series are not convergent in norm.
scientific article; zbMATH DE number 4202670