The operator e-tA and its trace Tr e-tA for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter-ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0+. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A - λ)-1. The main step consists in approximating this family by a parametrix of A - λ constructed within a suitable parameter-dependent calculus.
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