Bi-isometries reducing the hyper-ranges of the coordinates

Abstract

Let $(S_1,S_2)$ be a bi-isometry, that is, a pair of commuting isometries $S_1$ and $S_2$ on a complex Hilbert space $\mathcal{H}.$ By the von Neumann-Wold decomposition, the hyper-range ${\mathcal{H}}_{\infty }(S_1):={\cap }_{n=0}^{\infty }{S}_1^n\mathcal{H}$ of $S_1$ reduces $S_1$ to a unitary operator. Although ${\mathcal{H}}_{\infty }(S_1)$ is an invariant subspace for $S_2,$ in general, ${\mathcal{H}}_{\infty }(S_1)$ is not a reducing subspace for $S_2.$ We show that ${\mathcal{H}}_{\infty }(S_1)$ reduces $S_2$ to an isometry if and only if the subspaces $S_2(\ker S_1^{\ast })$ and ${\mathcal{H}}_{\infty }(S_1)$ of $\mathcal{H}$ are orthogonal. Further, we describe all bi-isometries $(S_1,S_2)$ satisfying the orthogonality condition mentioned above.

11 pages