On the number of eigenvalues of a matrix operator

We consider a matrix operator $H$ in the Fock space. We prove the finiteness of the number of negative eigenvalues of $H$ if the corresponding generalized Friedrichs model has the zero eigenvalue ($0=\min\sigma_\mathrm{ess}(H)$). We also prove that $H$ has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of $H$ below $z$ as $z\to-0$.