Exact g-frames in Hilbert spaces (Q601318)

Let \(U\) and \(V\) be two complex Hilbert spaces, let \(\{V_j\}_{j\in J}\) be a sequence of closed subspaces of \(V,\) where \(J\subset\mathbb Z,\) and let \(L(U,V_j)\) be the collection of all bounded linear spaces from \(U\) to \(V_j.\) For a \(g\)-frame \(\{\Lambda_j:\Lambda_j\in L(U,V_j)\}_{j\in J}\) for \(V\) with respect to \(\{V_j\}_{j\in J}\) (i.e. there exist \(A,B>0\) such that \[ A\| f\|^2\leq \sum_{j\in J}\| \Lambda_jf\|^2\leq B\| f\|^2 \] for all \(f\in V)\) which is exact (i.e. it ceases to be a \(g\)-frame whenever any one of its elements is removed), the authors find their characterization, an equivalent relation between them and \(g\)-Riesz bases under some conditions. Moreover they consider the stability of an exact \(g\)-frame under perturbation.