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

We consider an operator function F defined on the interval $\user2{}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}$ whose values are semibounded self-adjoint operators in the Hilbert space $\mathfrak{H}$ . To the operator function F we assign quantities $\mathcal{N}_\user1{F}$ and ν _F (λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ σ,τ]. We present conditions under which the estimate $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ holds. We also establish conditions for the relation $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.