  Frames have been investigated frequently over the last few decades due to
their valuable properties, which are desirable for various applications as well
as interesting for theory. Some applications additionally require distributed
processing techniques, which naturally leads to the concept of fusion frames
and fusion frame systems. The latter consists of a system of subspaces,
equipped with local frames on each of them, and a global frame.
  In this paper, we investigate the relations of the associated frame-related
operators on all those three levels. For that we provide a detailed
investigation on bounded block diagonal operators between Hilbert direct sums.
We give the relation of the frame-related operators of the fusion frame and the
corresponding frame systems in terms of operator identities. By applying these
identities we prove further properties of fusion frame systems.
