For operators belonging either to a class of global bisingularpseudodifferential operators on $R^m \times R^n$ or to a class of bisingularpseudodifferential operators on a product $M \times N$ of two closed smoothmanifolds, we show the equivalence of their ellipticity (defined by theinvertibility of certain associated homogeneous principal symbols) and theirFredholm mapping property in associated scales of Sobolev spaces. We also provethe spectral invariance of these operator classes and then extend these resultsto the even larger classes of Toeplitz type operators.
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机译：对于属于$ R ^ m \ times R ^ n $上的一类全局双单数伪微分算子或属于两个闭合光滑流形的乘积$ M \ times N $上的一类双单数伪微分算子，我们证明了其椭圆性的等价性（由某些关联的齐次主体符号的可逆性定义）及其在Sobolev空间的关联比例中的Fredholm映射属性。我们还证明了这些算子类别的谱不变性，然后将这些结果扩展到更大的Toeplitz类型算子类别。
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