Geometry of epimorphisms and frames

Abstract:

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