Let ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a smooth submanifold of the Hilbert space ${\mathcal{B}}_2\left(\mathcal{H}\right)$ 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 $N<\mathrm{\infty }$. Furthermore, ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a homogeneous space of ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}\times {\mathcal{U}}_{\left(\mathrm{\infty }\right)}$, where ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}$ 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...

We present new results related to various equivalents of the mixed-type reverse order law for the Moore-Penrose inverse for operators on Hilbert spaces. Recent finite-dimensional results of Tian are extended to Hilbert space operators.

Let t be a regular operator between Hilbert C*-modules and $t^{†}$ be its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of the Gram operator t*t. More precisely, we study some conditions ensuring that $t^{†}={(t*t)}^{†}t*=t*{(tt*)}^{†}$ and ${(t*t)}^{†}=t^{†}t{*}^{†}$. As an application, we get some results for densely defined closed operators on Hilbert C*-modules over C*-algebras of compact operators.