Description of commutants of integration operator in analytic spaces
Let D be a simply connected region in \({\mathbb{C}}\), starlike with respect to a fixed point \(\alpha\in {\mathbb{C}}\); let A(D) be the space of all analytic functions in D, with the topology of compact convergence, and \[ (J_{\alpha}f)(z)=\int^{z}_{\alpha}f(t)dt,\quad \forall f\in A(D),\quad z\in {\mathbb{C}}. \] The author proves that: T is a commutant of \(J_{\alpha}\) in A(D), i.e. \(TJ_{\alpha}=J_{\alpha}T\), if and only if \[ (1)\quad(Tf)(z)=\phi(\alpha)f(z)+\int^{z}_{\alpha}\phi '(z+\alpha -t)f(t)dt,\quad \forall f\in A(D), \] where \(\phi\in A(D).\) T is an isomorphism of A(D) and commutes with \(J_{\alpha}\) if and only if T satisfies (1) and \(\phi(\alpha)=(Ta)_{z=\alpha}\neq 0.\) When \(\alpha =0\), all the results of the paper are known.
scientific article; zbMATH DE number 3851843