Parameter-dependent Pseudodifferential Operators of Toelitz Type

We present a calculus of pseudodifferential operators that contains both usual parameter-dependent operators - where a real parameter au enters as an additional covariable - as well as operators not depending on au. Parameter-ellipticity is characterized by the invertibility of three associated principal symbols. The homogeneous principal symbol is not smooth on the whole co-sphere bundle but only admits directional limits at the north-poles, encoded by a principal angular symbol. Furthermore there is a limit-family for au o+infty. Ellipticity permits to construct parametrices that are inverses for large values of the parameter. We then obtain sub-calculi of Toeplitz type with a corresponding symbol structure. In particular, we discuss invertibility of operators of the form P_1A( au)P_0 where both P_0 and P_1 are zero-order projections and A( au) is a usual parameter-dependent operator of arbitrary order or A( au)= au^{mu}-A with a pseudodifferential operator A of positive integer order mu.