  We consider the periodic Jacobi operator $J$ with finitely supported
perturbations on the half-lattice. We describe all eigenvalues and resonances
of $J$ and give their properties. We solve the inverse resonance problem: we
prove that the mapping from finitely supported perturbations to the Jost
functions is one-to-one and onto, we show how the Jost functions can be
reconstructed from the eigenvalues, resonances and the set of zeros of
$S(\l)-1,$ where $S(\l)$ is the scattering matrix.
