An eigenvalue multiplicity formula for the Schur complement of a $3\times3$ block operator matrix

We consider the Schur complement $S(\lambda)$ with real spectral parameter $\lambda$ corresponding to a certain $3\times3$ block operator matrix. In our case the essential spectrum of $S(\lambda)$ can have gaps. We obtain formulas for the number and multiplicities of eigenvalues belonging to an arbitrary interval outside the essential spectrum of $S(\lambda)$.