Fredholm and semi-Fredholm composition operators on the small Bloch-type spaces
Let \(\mathbb D\) be the unit disk in the complex plane and let \(H({\mathbb D})\) be the space of all analytic functions on \(\mathbb D\). For a non-constant analytic function \(\varphi: {\mathbb D}\rightarrow {\mathbb D}\), the composition operator \(C_{\varphi}\) on the Banach space \({\mathcal X}\subset H({\mathbb D})\) is defined by \(C_{\varphi}f=f\circ\varphi\), \(f\in{\mathcal X}\). For \(\alpha>0\), the \(\alpha\)-Bloch space \({\mathcal B}^{\alpha}\) is the space of functions \(f\in H({\mathbb D})\) with the finite norm \[ \| f \|_{{\mathcal B}^{\alpha}} = |f(0)| + \sup_{z\in\mathbb D}((1-|z|^{2})^{\alpha}|f'(z)|). \] In the case \(0<\alpha<1\), these spaces are called the small Bloch-type spaces. The main result of the paper is as follows. The composition operator \(C_{\varphi}\) on the small Bloch-type space \({\mathcal B}_{\alpha}\), \(0<\alpha<1\), is Fredholm if and only if \(\varphi\) is a disk automorphism (i.e., if and only if \(C_{\varphi}\) is invertible). Under some additional assumptions, the authors find necessary and sufficient conditions so that \(C_{\varphi}\) has closed range on \({\mathcal B}^{\alpha}\), \(0<\alpha<1\).
scientific article; zbMATH DE number 6160913
scientific article