Pseudodifferential operators on Hilbert bundles

Abstract

makes sense when u is a Hilbert space-valued function and p is operator valued. The theory is developed with the aim of discovering a natural class of “elliptic” operators, i.e., some generalization of the usual elliptic operators, retaining the Fredholm property. We investigate those symbols which are themselves Fredholm operator valued and invertible near infinity because it is clear that they already possess a topological index as defined by Atiyah and Singer [3]. A Fredholm operator is invertible modulo compact operators and the first step towards finding Fredholm operators is to find a relevant class of compact operators. We show that compact operator-valued symbols of order 03 give rise to compact operators and may hence take the place of smoothing operators in our theory. Then we carry out Hormander’s calculus for constructing parametrices. The first interesting example arises as follows. Let