Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions

We consider an operator function $F$ defined on the interval $[\sigma,\tau]\subset \mathbb R$ whose values are semibounded self-adjoint operators in the Hilbert space $\mathfrak H$. To the operator function $F$ we assign quantities $\mathscr N_F$ and $\nu_F(\lambda)$ that are, respectively, the number of eigenvalues of the operator function $F$ on the half-interval $[\sigma,\tau)$ and the number of negative eigenvalues of the operator $F(\lambda)$ for an arbitrary $\lambda\in[\sigma,\tau]$. We present conditions under which the estimate $\mathscr N_F\geqslant\nu_F(\tau)-\nu_F(\sigma)$ holds. We also establish conditions for the relation $\mathscr N_F=\nu_F(\tau)-\nu_F(\sigma)$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.