  We compute non-perturbative gluon spectral functions at finite temperature in
quenched QCD with the maximum entropy method. We also provide a closed loop
equation for the spectral function of the energy-momentum tensor in terms of
the gluon spectral function. This setup is then used for computing the shear
viscosity over entropy ratio $\eta/s$ in a temperature range from about $0.4\,
T_c$ to $4.5\, T_c$. The ratio $\eta /s$ has a minimum at about $1.25\, T_c$
with the value of about 0.115. We also discuss extensions of the present
results to QCD.
