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

Abstract

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