Topology and smooth structure of pseudoframes

ESTEBAN ANDRUCHOW; JORGE ANTEZANA; GUSTAVO CORACH

INTEGRAL EQUATIONS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Lugar: BASEL; 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, CF_S of  commutative pseudo-frames and X_S 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}, {h_n}, F:l^2--> H,  F({c_n})=sum_n c_n f_n, H:l^2 --> H,  H({c_n})=sum_n c_n h_n. We prove that, with this identification, the sets F_S, CF_S and X_S are complemented submanifolds of B(l^2,H)xB(l^2,H). We examine in more detail X_S, which carries a locally transitive action from the general linear group GL(l^2). For instance, we characterize the homotopy theory of X_S and we prove that X_S is a strong deformation retract both of F_S and CF_S; therefore these sets share many of their topological properties.