Topology and smooth structure for pseudoframes

E. ANDRUCHOW, J. ANTEZANA, G. CORACH

INTEGRAL EQUATIONS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Año: 2010 vol. 67 p. 451 - 466

0378-620X

Given a closed subspace $\s$ of a Hilbert space $\h$, we study the sets $\f_\s$ of pseudo-frames, $\c\f_\s$ of  commutative pseudo-frames and $\sub{\ekis}{\ese}$ 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 $(\{f_n\}_{n\in\zN},\{h_n\}_{n\in\zN})$,$$F:\ell^2\to \hil, \ \ F\big(\{c_n\}_{n\in\N}\big)=\sum_n c_n f_n,$$and$$H:\ell^2 \to \hil, \ \  H=(\{c_n\}_{n\in\N})=\sum_n c_n h_n.$$We prove that, with this identification, the sets $\f_\s$, $\c\f_\s$ and $\sub{\ekis}{\ese}$ are complemented submanifolds of $\b(\ell^2,\h)\times \b(\ell^2,\h)$. We examine in more detail $\sub{\ekis}{\ese}$, which carries a locally transitive action from the general linear group $GL(\ell^2)$. For instance, we characterize the homotopy theory of $\sub{\ekis}{\ese}$ and we prove that $\sub{\ekis}{\ese}$ is a strong deformation retract both of $\f_\s$ and $\c\f_\s$; therefore these sets share many of their topological properties.}