On the sum of generalized frames in Hilbert spaces (Q2050121)

The paper deals with \(K\)-frames, \(K\)-\(g\)-frames and controlled frames in Hilbert spaces. The main results are the following. \par i) Let \(\{f_n\}\) be a Bessel sequence of a Hilbert space \(\mathcal{H}\) and \(\mathcal{KF}\) the set of operators \(K\in \mathcal{B}(\mathcal{H})\) such that \(\{f_n\}\) is a \(K\)-frame. Then \(\mathcal{KF}\) is a right ideal (but not necessarily a left ideal) of \(\mathcal{B}(\mathcal{H})\). \par ii) Sufficient conditions for the finite sum of \(K\)-frames to be a \(K\)-frame. \par iii) Results of the same type about \(K\)-\(g\)-frames and controlled frames. The paper contains also some examples related to the results.