  We study heat traces associated with positive unbounded operators with
compact inverses. With the help of the inverse Mellin transform we derive
necessary conditions for the existence of a short time asymptotic expansion.
The conditions are formulated in terms of the meromorphic extension of the
associated spectral zeta-functions and proven to be verified for a large class
of operators. We also address the problem of convergence of the obtained
asymptotic expansions. General results are illustrated with a number of
explicit examples.
