New proof of Nagnibida's theorem (Q2369473)

Let \(\Omega \) be a simple-connected domain in the plane, \(\alpha \in \Omega \). Assume that \(\Omega \) is star-shaped with respect to \(\alpha \). Let \(\text{Hol}(\Omega )\) be the space of all functions holomorphic in \(\Omega \). Set \(I^p = \{ f\in \text{Hol}(\Omega ): f(\alpha )=f'( \alpha )=\dots =f^{(p-1)}(\alpha )=0\} \). Let \(J_\alpha \) denote the integration operator acting in \(\text{Hol} (\Omega )\) by the formula \((J_\alpha f)(z)= \int_\alpha ^z f(t)\,dt\), where the integral is taken over the line segment with the ends \(\alpha \) and \(z\). The paper under review gives a new proof of Nagnibida's theorem which says that the lattice of \(J_\alpha \) invariant subspaces of \(\text{Hol}(\Omega )\) is equal to \(\{ I^p:p\geq 1\}\).