The density theorem and its applications for operator-valued Gabor frames on LCA groups

In this paper, we introduce and study the operator-valued Gabor frames on locally compact abelian groups. We first prove a density theorem which says that for an operator-valued Gabor frame, the index subgroup is co-compact if and only if the window operator is a Hilbert-Schmidt operator. From this result, we obtain a necessary condition for two operator-valued Gabor frames to be dual. Then, some characterizations of operator-valued Gabor frames are given and especially, we show that operator-valued Gabor frames share similar properties with ordinary Gabor frames. Finally, the concept of short-time Fourier transforms for Hilbert-Schmidt operators is introduced to prove that an operator-valued Gabor system indexed by the entire phase-space is Bessel but not a frame for the space of Hilbert-Schmidt operators. Even in the background of Euclidean spaces, our results are also original.