Commutative properties of invertible multipliers in relation to representation of their inverses

Recently frame multipliers and in particular their invertibility became an important subject of research. In particular, it was shown that given an invertible frame multiplier, a particular dual frame can be determined, such that the inverse is again a multiplier, using this dual sequence. In the paper we deal with the question if these particular duals keep the specific structure of the primal frames. We investigate pseudo-coherent frames and summarize results showing that the structure is kept for frame-like operators, i.e. multipliers with constant symbols. For more general symbols we show result that under additional assumptions, in particular a commutation relation, the symbol being constant is a necessary condition. We reformulate the results for the particular case of Gabor systems.