On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions (Q1018173)

Let \(H\) be a Hilbert space of analytic functions on the unit disc with \(\|M_n\|\leq1\), where \(M_z\) denotes the operator of multiplication by the identity function on the unit disc. Under certain conditions on \(H\), it has been shown by \textit{A.\,Aleman, S.\,Richter} and \textit{C.\,Sundberg} [Trans.\ Am.\ Math.\ Soc.\ 359, No.\,7, 3369--3407 (2007; Zbl 1126.47023)] that all invariant subspaces have index 1 if and only if \(\lim_{k\to\infty}\|M^k_zf\|\not=0\) for all \(f\in H\) with \(f\not\equiv 0\). The author gives an example to show that the above theorem is not true for the Hilbert spaces of \(C^n\)-valued analytic functions. Also, a correct generalization of the theorem above is proved.