High-resolution Fourier transform scanning tunneling spectroscopy (FT-STS) is used to study many-body effects on the surface state of Ag(111). Our results reveal a kink in the otherwise parabolic band dispersion of the surface electrons and an increase in the quasiparticle lifetime near the Fermi energy Ef. The experimental data are accurately modeled with the T-matrix formalism for scattering from a single impurity, assuming that the surface electrons are dressed by the electron-electron and electron-phonon interactions. We confirm the latter as the interaction responsible for the deviations from bare dispersion. We further show how FT-STS can be used to simultaneously extract real and imaginary parts of the self-energy for both occupied and unoccupied states with a momentum and energy resolution competitive with angle-resolved photoemission spectroscopy. From our quantitative analysis of the data we extract a Debye energy of ℏΩD=14±1 meV and an electron-phonon coupling strength of λ=0.13±0.02, consistent with previous results. This proof-of-principle measurement advances FT-STS as a method for probing many body effects, which give rise to a rich array of material properties.