Topology and Smooth Structure for Pseudoframes

Abstract

Given a closed subspace S of a Hilbert space H, we study the sets FS of pseudo-frames, CFS of commutative pseudo-frames and XS of dual frames for S, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({fn}n∈N, {hn}n∈N), F : ! 2 → H, F ({cn}n∈N) = ! n cnfn, and H : ! 2 → H, H = ({cn}n∈N) = ! n cnhn. We prove that, with this identification, the sets FS , CFS and XS are complemented submanifolds of B(!2, H)×B(!2, H). We examine in more detail XS , which carries a locally transitive action from the general linear group GL(! 2). For instance, we characterize the homotopy theory of XS and we prove that XS is a strong deformation retract both of FS and CFS; therefore these sets share many of their topological properties.