  We introduce an operator valued Short-Time Fourier Transform for certain
classes of operators with operator windows, and show that the transform acts in
an analogous way to the Short-Time Fourier Transform for functions, in
particular giving rise to a family of vector-valued reproducing kernel Banach
spaces, the so called coorbit spaces, as spaces of operators. As a result of
this structure the operators generating equivalent norms on the function
modulation spaces are fully classified. We show that these operator spaces have
the same atomic decomposition properties as the function spaces, and use this
to give a characterisation of the spaces using localisation operators.
