Analysis of the discrete spectrum of the family of $3\times 3$ operator matrices

We consider the family of $3\times 3$ operator matrices $\mathbf{H}(K),$ $K \in {\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $\Lambda \subset {\Bbb T}^3$toprovetheexistenceofinfinitelymanyeigenvaluesof${\bf H}(K)$forall$K \in \Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K\in \mathrm{\Lambda },$ the number $N(K, z)$ofeigenvaluesof${\bf H}(K)$lyingontheleftof$z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$with$0<{\mathcal U}_0<\infty,$ independently on the cardinality of $\mathrm{\Lambda }\mathrm{.}$ Moreover, we prove that for any $K\in \mathrm{\Lambda }$ the operator $\mathbf{H}(K)$ has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.