Generalized frames and their redundancy (Q2701593)

In this paper the authors prove that continuously indexed frames in separable Hilbert spaces with no redundancy (or excess) are equivalent to discretely indexed sets. The result applies to general not necessarily separated Hilbert spaces, by restricting the analysis to closed countably generated subspaces. More specific, let \(H\) be a Hilbert space and \((M,S,\mu)\) a measure space. Then a generalized frame in \(H\) indexed by \(M\) is a family \(h=\{h_m\in H\); \(m\in M\}\) such that: (a) \(\forall f\in H\), \(m\mapsto Tf(m):= \langle h_m,f\rangle\) is measurable; (b) there are \(0<A,B<\infty\) such that \(\forall f\in H\), \(A\|f{\|}_H^2 \leq \|Tf{\|}_{L^2(M;\mu)}^2\leq B\|f{\|}_H^2\). Recall also a measurable subset \(E\) of \(H\) is called an atom if \(0<\mu(E)<\infty\) and \(E\) contains no measurable subset \(F\) such that \(0<\mu(F)<\mu(E)\). The main result reads (the reviewer takes the liberty to fix a typo): NEWLINENEWLINENEWLINETheorem 2.2: Let \(h\) be a generalized frame in \(H\) indexed by \((M,S,\mu)\) and assume \(Im T=L^2(M,d\mu)\). Then for every countable subset \(L\) of \(H\), there exists a countable collection \(\{ E_i;i\in\Lambda\}\) of disjoint measurable sets such that \(\widetilde{f}=\sum c_{fi}\chi_i\) for all \(f\) in the closed linear span of \(L\), where \(\{c_{fi};i\in\Lambda\}\) is a set of complex numbers depending on \(f\), and \(\chi_i\) denotes the characteristic function of \(E_i\). In particular, if \(H\) is an infinite dimensional separable space, then \(L^2(M;\mu)\) is isometrically isomorphic to the weighted space \(l^2_w\) consisting of all sequences \(\{c_i\}\) with \(\|\{c_i\} {\|}^2=\sum_i |c_i|^2w_i<\infty\), where \(w_i=\mu(E_i)\) for a fixed collection of disjoint \(\mu\)-atoms \(\{E_1,E_2,\ldots\}\).