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Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert spaceand let $\mathcal{J}$ be a two-sided ideal of the algebra of bounded operators$\mathcal{B}(\mathcal{H})$. The groups $\mathcal{G} \ell_\mathcal{J}$ and$\mathcal{U}_{\mathcal{J}}$ consist of all the invertible operators and unitaryoperators of the form $I + \mathcal{J}$, respectively. We study the actions ofthese groups on the set of closed range operators. First, we find equivalentcharacterizations of the $\mathcal{G} \ell_\mathcal{J}$-orbits involving theessential codimension. These characterizations can be made more explicit in thecase of arithmetic mean closed ideals. Second, we give characterizations of the$\mathcal{U}_{\mathcal{J}}$-orbits by using recent results on restricteddiagonalization. Finally we introduce the notion of $\mathcal{J}$-equivalenceand $\mathcal{J}$-unitary equivalence between frames for subspaces of a Hilbertspace, and we apply our abstract results to obtain several results regardingduality and symmetric approximation of $\mathcal{J}$-equivalent frames.

Restricted orbits of closed range operators and equivalences between frames for subspaces