I am working in the area of graph theory, where the matrix $L$ is the combinatorial matrix of a graph, a positive semidefinite matrix.
The resolvent of a matrix $L$ (improperly called Green function) is defined as:
$$R(z)=(zI - L)^{-1}$$
In my calculations, related to a problem on graphs, I need to make sense of an expression that seems related to the resolvent, but with opposite sign. In particular the quantity that I am trying to make sense of is the following:
$$F(z)=\mathrm{Tr}\left \lbrack z(zI+L)^{-1} \right \rbrack$$
where $\mathrm{Tr}$ is the trace of a matrix. This quantity seems related in some way to the resolvent, but I see a positive sign in front of $L$, and the factor $z$ looks like computing the first momentum of a spectral density.
Is this possible to put the quantity $F$ in relation to the resolvent $R$? This would greatly help me to understand a lot of things.