On perturbation of local atoms for subspaces (Q2825732)

Let \(H\) be a Hilbert space and let \(H_0\) be a closed subspace of \(H\). A sequence \(\{f_i\}_{i\in I}\subset H\) is called a family of local atoms for \(H_0\), ifNEWLINENEWLINE(i) \(\{f_i\}_{i\in I}\) is a Bessel sequence in \(H\), i.e., there exists a real number \(B>0\) such that NEWLINE\[NEWLINE\sum_{i\in I}|\langle f,f_i\rangle|^2\leq B\| f\|^2NEWLINE\]NEWLINE for each \(f\in H\).NEWLINENEWLINE(ii) There exists a sequence of linear functionals \(\{c_i\}_{i\in I}\) and a real number \(C>0\) such that NEWLINE\[NEWLINE\| c_i(f)\|_{\ell^2}^2\leq C\| f\|^2NEWLINE\]NEWLINE for each \(f\in H_0\) and \(f=\sum_{i\in I}c_i(f) f_i\) for each \(f\in H_0\).NEWLINENEWLINEThe authors of the present paper obtain some results for perturbations of a family of local atoms, especially they show that under some conditions a family of local atoms is stable under perturbations. They also obtain some results for K-frames.