  Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space
and 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 unitary
operators of the form $I + \mathcal{J}$, respectively. We study the actions of
these groups on the set of closed range operators. First, we find equivalent
characterizations of the $\mathcal{G} \ell_\mathcal{J}$-orbits involving the
essential codimension. These characterizations can be made more explicit in the
case of arithmetic mean closed ideals. Second, we give characterizations of the
$\mathcal{U}_{\mathcal{J}}$-orbits by using recent results on restricted
diagonalization. Finally we introduce the notion of $\mathcal{J}$-equivalence
and $\mathcal{J}$-unitary equivalence between frames for subspaces of a Hilbert
space, and we apply our abstract results to obtain several results regarding
duality and symmetric approximation of $\mathcal{J}$-equivalent frames.
