I'm afraid the question below might turn out to be very stupid -
I just don't know how to make sense of two asymptotic expansions,
given the heat operator $e^{-t\triangle}$ with $\triangle$ a Laplace type operator on smooth functions over an n-dimensional manifold we have an asymptotic expansion of the kernel \begin{equation} \text{k}(e^{-t\triangle}) \backsim \sum_{j \geq 0} c_{\frac{-n - j}{2}} \,t^{\frac{-n - j}{2}} \end{equation} and we also have an asymptotic expansion for the trace, \begin{equation} \text{Tr}(e^{-t\triangle}) \backsim \sum_{j \geq 0} c_{\frac{-n - j}{2}} \,t^{\frac{-n - j}{2}} \end{equation}
Now, where is the difference in these two expansions ?
I realize this question suggests I don't know enough about both the kernel and the definition of the trace - which is absolutely true. The problem though is I don't know how to find an answer since all sources I have go on a large detour on how to construct the heat kernel, and on general definitions and properties of trace structures on operator algebras. I am totally aware I need to eventually understand these concepts, what I hope for with this question is to get a first idea in order to proceed. If that's not possible I'd also appreciate a reference to literature that starts at a low level so that I can get an overview.
Hope this makes sense, thanks a lot for your help!