Tight \(K\)-\(g\)-frame and its novel characterizations via atomic systems (Q1796544)

The authors characterize tight \(K\)-\(g\)-frames, a generalization of ordinary frames introduced by \textit{Y. Zhou} and \textit{Y. Zhu} [Acta Math. Sin., Chin. Ser. 56, No. 5, 799--806 (2013; Zbl 1299.42109)]. Although it is mentioned that this has been done via atomic systems, there is no explicit definition of atomic systems and their connection with this class of frames. The authors also study some properties of \(K\)-\(g\)-frames. In my view, some of the results are so obvious that there is no need for a proof, but the proofs are accurate and some helpful examples are included. Specific comments: (1) On page 2 Lemma 9, in condition (2) it would be better to mention that \(\{\Gamma_j\}_{j\in J}\) is the pointed \(g\)-Bessel sequence. (2) On page 6 the line number 5 from the top on the left hand, \(K^\ast f\in H\) should be changed to \(K_1^\ast f\in H\) and also all three lines 5 to 7 are not a reason for \(K_1^\ast f\) to belong to \(H\). In fact it belongs to the Hilbert space, because \(K_1^\ast\in L(H)\). But the explanation could be used for relation (36).