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.
The spectral function in physics tells you the probability that a particle with a certain momentum $k$ has a specific energy $\omega$. In other words, due to the Heisenberg uncertainty relation, a particle can have an energy that is distributed around a mean. The spectral function $A(k,\omega)$ describes exactly this distribution.
Therefore, in physics $\lambda(S)$ should translate into a vector that contains all the Eigenenergies of the respective system. Then the function $f(\lambda)$ should denote the distribution of energies a particle may have.
The answer given by Lagerbaer is most correct from a spectroscopist's point of view. One always measures a spectrum of energies. And the reason why one sees a spectrum of energies, is related to the uncertainty of the energy of a particle in a specific state. This distribution is just captured by the spectral function $A(k,\omega)$.