Wandering subspaces and quasi-wandering subspaces in the Bergman space (Q554290)

This paper gives a summary of papers by \textit{K. J. Izuchi}, \textit{K. H. Izuchi} and \textit{Y. Izuchi} [Integral Equations Oper. Theory 67, No. 2, 151--161 (2010; Zbl 1219.47016)] and [Arch. Math. 95, No. 5, 439--446 (2010; Zbl 1219.47017)]. However, it seems that the following result is not included in the cited papers. Let \(T\) be a contraction on a Hilbert space \(H\) which is bounded below. Assume that \(\| Tx\|^2+\| {T^*}2^Tx\|^2 \leq 2\| T^* T x\|^2\) and \(\| {T^*}^kx\| \to 0\) as \(k\to \infty\), for every \(x\in H\). Then \(H\) is the smallest closed \(T\)-invariant subspace which contains the orthogonal complement of the image of \(T\). An application of this result gives a simple proof of a Beurling type theorem for the Bergman shift on \(L_{a}^{2}\).