Geometry of epimorphisms and frames

Abstract

Using a bijection between the set B H \mathcal {B}_{\mathcal {H}} of all Bessel sequences in a (separable) Hilbert space H \mathcal {H} and the space L ( ℓ 2 , H ) L(\ell ^2 , \mathcal {H}) of all (bounded linear) operators from ℓ 2 \ell ^2 to H \mathcal {H} , we endow the set F \mathcal {F} of all frames in H \mathcal {H} with a natural topology for which we determine the connected components of F \mathcal {F} . We show that each component is a homogeneous space of the group G L ( ℓ 2 ) GL( \ell ^2) of invertible operators of ℓ 2 \ell ^2 . This geometrical result shows that every smooth curve in F \mathcal {F} can be lifted to a curve in G L ( ℓ 2 ) GL( \ell ^2) : given a smooth curve γ \gamma in F \mathcal {F} such that γ ( 0 ) = Ξ \gamma (0)= \Xi , there exists a smooth curve Γ \Gamma in G L ( ℓ 2 ) GL(\ell ^2) such that γ = Γ ⋅ Ξ \gamma = \Gamma \cdot \Xi , where the dot indicates the action of G L ( ℓ 2 ) GL( \ell ^2) over F \mathcal {F} . We also present a similar study of the set of Riesz sequences.