Geometry of epimorphisms and frames

G. CORACH, M. PACHECO Y D. STOJANOFF

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMS

Año: 2004 vol. 132 p. 2039 - 2049

0002-9939

Using a bijection between the set Bess(H) of all Bessel sequences in a (separable)  Hilbert space H and the space L( l_2 , H) of all (bounded linear) operators from l_2 to H, we endow the set F of all frames in H with a natural topology for which we  determine the connected components of F.  We show that each component is a homogeneous space of the group G( l_2) of invertible operators of  l_2. This geometrical result shows that every smooth curve in F can be lift to a curve in G( l_2): given a smooth curve $\gamma$ in F such that  $\gamma (0)= Xi$, there exists a smooth curve $\Gamma$ in G( l_2) such that $\gamma = \Gamma \cdot Xi$, where the dot indicates the action of G( l_2) over F.We also present a similar study of the set of Riesz sequences.