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To achieve our main research goal, first we survey the approaches towards dual fusion frames existing in the literature and agree on the notion of duality for fusion frames in the sense of Kutyniok, Paternostro and Philipp (Oper. Matrices 11 (2017), no. 2, 301-336). As a main result we show that different fusion frames have different dual fusion frames. Moreover, this duality notion leads to a new definition of Bessel fusion multipliers which is a slightly modified version of the commonly used definitions. Particularly, we show that with this definition in many cases Bessel fusion multipliers behave similar to ordinary Bessel multipliers. Finally, special attention is devoted to the study of dual fusion frames induced by an invertible Bessel fusion multiplier.

In this paper, we consider oblique dual frames and particularly, we obtain some of their characterizations. Among other things, special attention is devoted to the study of the effect of perturbations of frame sequences on their oblique duals. In particular, as a surprising result, we show that a frame sequence is uniquely determined by the set of its oblique dual.