Markus J. Pflaum
The Normal Symbol on Riemannian Manifolds
||August 31, 1998
||pseudodifferential operators on manifolds, asymptotic expansions, symbol calculus
For an arbitrary Riemannian manifold X
and Hermitian vector bundles E and F over X we define the
notion of the normal symbol of a pseudodifferential operator P
from E to F.
The normal symbol of P is a certain smooth function from the
cotangent bundle T*X to the homomorphism bundle Hom (E,F)
and depends on the metric structures, resp. the corresponding
connections on X, E and F.
It is shown that by a natural integral formula the
pseudodifferential operator P can be recovered from its symbol.
Thus, modulo smoothing operators, resp. smoothing symbols, we
receive a linear bijective correspondence between the space of
symbols and the space of pseudodifferential operators on X.
This correspondence comprises a natural transformation between
A formula for the asymptotic expansion of the product symbol
of two pseudodifferential operators in terms of the symbols of its
factors is given. Furthermore an expression for the symbol of the
adjoint is derived.
Finally the question of invertibility of pseudodifferential operators
is considered. For that we use the normal symbol to establish a new
and general notion of elliptic pseudodifferential operators on
Humboldt Universität zu Berlin, Mathematisches Institut, Unter den Linden 6, 10099 Berlin