Let M be a compact
Riemannian manifold with smooth boundary ∂M. We study the asymptotic
expansions associated with the generalized heat operator Qe−tPℬ with suitable
boundary conditions. A new invariant defined on the boundary of M is introduced,
and a method is given that relates the heat content asymptotics for the generalized
heat operator and the standard heat operator e−tPℬ with the new boundary
asymptotics. As an application, we compute the boundary asymptotics associated
with an operator of Laplace type, and the asymptotics for a generalized operator
constructed from an operator of Dirac type.
