  For operators belonging either to a class of global bisingular
pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular
pseudodifferential operators on a product $M \times N$ of two closed smooth
manifolds, we show the equivalence of their ellipticity (defined by the
invertibility of certain associated homogeneous principal symbols) and their
Fredholm mapping property in associated scales of Sobolev spaces. We also prove
the spectral invariance of these operator classes and then extend these results
to the even larger classes of Toeplitz type operators.
