Stability of quasiframes with quasiframe operators (Q2765942)

Let \(\{\rho_n\}_{n\in N}\) be a sequence in \(\ell^2(N)\). The authors call a sequence \(\{x_n\}\) in a Hilbert space \(H\) a quasiframe for \(H\) if there exist constants \(A,B>0\) such that NEWLINE\[NEWLINEA\|x\|^2 \leq \sum_{n\in N}|\rho_n \langle x,x_n\rangle|^2 \leq B\|x\|^2, \;\forall x\in H.NEWLINE\]NEWLINE Note that if \(\{x_n\}\) is a quasiframe, then \(\{\overline{\rho_n}x_n\}\) is a frame in the traditional sense. Section 2 of the present paper restates known frame results in terms of quasiframes, and Section 3 gives applications to vector-valued measures.