Parameter-dependent Pseudodifferential Operators of Toeplitz Type on Closed Manifolds

We present a calculus of zero-order parameter-dependent pseudodifferential operators on a closed manifold $M$ that contains both usual parameter-dependent operators - where the parameter enters as an additional covariable - as well as operators independent of the parameter. Parameter-ellipticity is characterized by the invertibility of three associated principal symbols. In case of ellipticity we can construct a parametrix that is an inverse for large values of the parameter. We then extend this parametrix-construction to operators of Toeplitz type, in particular, to operators of the form $P_1A(\tau)P_0$ where both $P_0$ and $P_1$ are zero-order projections and $A(\tau)$ is a usual parameter-dependent operator of arbitrary order or $A(\tau)=\tau^{\mu}-A$ with a pseudodifferential operator $A$ of positiv order $\mu\in\nz$ $($in this case $P_0=P_1)$.