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

TL;DR

This paper investigates the actions of specific operator groups on closed range operators in Hilbert spaces, characterizes their orbits, and explores equivalences between frames for subspaces, with applications to duality and symmetric approximation.

Contribution

It provides new characterizations of group orbits on operators using essential codimension and recent diagonalization results, and introduces $\

Findings

Characterizations of $\

Applications to frame duality and approximation

Abstract

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.