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

CHIUMIENTO, EDUARDO; MASSEY, PEDRO

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Año: 2023

0039-3223

Let $cH$ be a separable infinite-dimensional complex Hilbert space and let $cJ$ be a two-sided ideal of the algebra of bounded operators $cB(cH)$. The groups $cG ell_cJ$ and $cU_cJ$ consist of all the invertible operators and unitary operators of the form $I + cJ$, respectively. We study the actions of these groups on the set of closed range operators. First, we find equivalent characterizations of the $cG ell_cJ$-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 $cU_cJ$-orbits by using recent results on restricted diagonalization. Finally we introduce the notion of $cJ$-equivalence and $cJ$-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 $cJ$-equivalent frames.