Bi-isometries reducing the hyper-ranges of the coordinates

TL;DR

This paper investigates conditions under which the hyper-range of one isometry in a bi-isometry pair reduces the second isometry to an isometry, providing a characterization and description of such pairs.

Contribution

It establishes a necessary and sufficient orthogonality condition for the hyper-range of one isometry to reduce the other to an isometry in bi-isometries.

Findings

Hyper-range reduces the second isometry to an isometry if and only if certain subspaces are orthogonal.

Provides a complete description of bi-isometries satisfying the orthogonality condition.

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 $\mathscr H.$ By the von Neumann-Wold decomposition, the hyper-range $\mathscr H_\infty(S_1):=\cap_{n=0}^\infty S^n_1\mathscr H$ of $S_1$ reduces $S_1$ to a unitary operator. Although $\mathscr H_\infty(S_1)$ is an invariant subspace for $S_2,$ in general, $\mathscr H_\infty(S_1)$ is not a reducing subspace for $S_2.$ We show that $\mathscr H_\infty(S_1)$ reduces $S_2$ to an isometry if and only if the subspaces $S_2(\ker S^*_1)$ and $\mathscr H_\infty(S_1)$ of $\mathscr H$ are orthogonal. Further, we describe all bi-isometries $(S_1, S_2)$ satisfying the orthogonality condition mentioned above.