  Starting out from a new description of a class of parameter-dependent
pseudodifferential operators with finite regularity number due to G. Grubb, we
introduce a calculus of parameter-dependent, poly-homogeneous symbols whose
homogeneous components have a particular type of point-singularity in the
covariable-parameter space. Such symbols admit intrinsically a second kind of
expansion which is closely related to the expansion in the Grubb-Seeley
calculus and permits to recover the resolvent-trace expansion for elliptic
pseudodifferential oerators originally proved by Grubb-Seeley. Another
application is the invertibility of parameter-dependent operators of Toeplitz
type, i.e., operators acting in subspaces determined by zero-order
pseudodifferential idempotents.
