Reproducing formulas for operator-valued Gabor frames on LCA groups

In this article, we investigate the reproducing formulas of operator-valued Gabor frames on an LCA group G with time-frequency shifts along a closed subgroup of the phase space. We first determine conditions on generators of such operator-valued systems that provide reproducing formulas of functions in $L^2\left(G\right)$. Then the weak density theorem, the Janssen representation and the Bessel duality principle for operator-valued Gabor systems are obtained. These extend well-established results on classical Gabor systems. While our partial results are similar to one of Gabor g-frames on $R^d$, introduced by E. Skrettingland [29], our methods are completely different, which provide new ideas of studying OPV-Gabor frames.