Please, I would like to understand why you call the function $A(k,\omega)$ (here :The Spectral Function in Many-Body Physics and its Relation to Quasiparticles ) a spectral function? For me, as a mathematician, a spectral function is a function which writes : $F(S)=f( \lambda(S))$ where for example $S$ is a symmetric matrix and $\lambda(S)$ is the vector of eigenvalues of $S$. Thank you in advance, Sincerely.
If you perform spectroscopy on a material (be it angular resolved photoemission spectroscopy (ARPES) or scanning tunneling spectroscopy (STS) or whatever method you fancy), the quantity you measure is roughly related to $A(k,\omega)$ (with additional prefactors and matrix elements depending on your method of choice.
Thus, performing spectroscopy on a sample provides you with information on $A(k,\omega)$, and hence we call it the spectral function.