Infiniteness of the number of eigenvalues embedded in the essential spectrum of a $2\times2$ operator matrix

In the present paper a $2\times2$ block operator matrix $\mathbf H$ is considered as a bounded self-adjoint operator in the direct sum of two Hilbert spaces. The structure of the essential spectrum of $\mathbf H$ is studied. Under some natural conditions the infiniteness of the number of eigenvalues is proved, located inside, in the gap or below the bottom of the essential spectrum of $\mathbf H$.