Restricted orbits of closed range operators and equivalences between frames for subspaces
Volume 276 / 2024
Abstract
Let $\mathcal {H}$ be a separable infinite-dimensional complex Hilbert space and let $\mathcal {J}$ be a two-sided ideal of the algebra $\mathcal {B}(\mathcal {H})$ of bounded operators. The groups $\mathcal {G}\ell _{\mathcal {J}}$ and $\mathcal {U}_{\mathcal {J}}$ consist of all the invertible and unitary operators of the form $I + \mathcal {J}$, respectively. We study several 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.
