Metrics in the set of partial isometries with finite rank

Andruchow, Esteban ; Corach, Gustavo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 1, pp. 31-44

Résumé

Let $\mathcal{I}_{(\infty)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that $\mathcal{I}_{(\infty)}$ is a smooth submanifold of the Hilbert space $\mathcal{B}_{2}(\mathcal{H})$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set $\mathcal{I}_{N}$, which consists of all partial isometries of rank $N \infty$. Furthermore, $\mathcal{I}_{(\infty)}$ is a homogeneous space of $\mathcal{U}_{(\infty)} \times \mathcal{U}_{(\infty)}$, where $\mathcal{U}_{(\infty)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics in $\mathcal{I}_{(\infty)}$: one, via the ambient inner product of $\mathcal{B}_{2}(\mathcal{H})$, the other, by means of the group action. We show that both metrics are equivalent and complete.