The two-dimensional Holstein model is studied by means of direct Lancz
os diagonalization preserving the full dynamics and quantum nature of
phonons. We present numerical exact results for the single-particle sp
ectral function. the polaronic quasiparticle weight, and the optical c
onductivity. The polaron band dispersion is derived both from exact di
agonalization of small lattices and analytic calculation of the polaro
n self-energy.