g-\((M,\mu)\)-frames in Hilbert spaces (Q2905986)

This paper deals with a notion of g-\((M,\mu)\)-frames in Hilbert spaces. Let \((M,\mu)\) be a measure space, \(U\) and \(V\) be two Hilbert spaces. \(\{V_m\}_{m\in M}\) is a family of subspaces of \(V\), and \(L(U,V_m)\) is the set of all linear bounded operators from \(U\) to \(V_m\). A family \(\{\Lambda_m\in L(U,V_m):\;m\in M\}\) is called a g-\((M,\mu)\)-frame for \(U\) with respect to \(\{V_m\}_{m\in M}\) if there exist positive constants \(A\leq B\) such that, for all \(f\in U\), NEWLINE\[NEWLINE A\|f\|^2_{U}\leq \int_M \|\Lambda_m f\|^2_V \,d\mu(m) \leq B\|f\|^2_U. NEWLINE\]NEWLINE This notion of g-\((M,\mu)\)-frames generalizes the concepts of frames, pseudoframes, g-frames and also \((M,\mu)\)-frames. Some equivalent characterizations and a dual property of g-\((M,\mu)\)-frames are then presented. Finally, the author extends some known results on the perturbation of g-frames and also \((M,\mu)\)-frames to the scale of g-\((M,\mu)\)-frames.