Invariant subspaces of parabolic self-maps in the Hardy space (Q2275680)

Let \(C_\varphi\) be the composition operator on the Hardy space \({\mathcal H}_2\), where \(\varphi\) is the map of the unit disk \(\mathbb D\) in the complex plane into itself, defined by the formula \[ \varphi_ a(z)=\frac{(2-a)z+a}{-az+2+a}, \qquad z\in {\mathbb D}, \] where \(\Re a>0\). It is proved that every invariant subspace of this operator is the closed span of some set of its eigenfunctions. As a consequence, it follows that such operators have no nontrivial reducing subspaces.