2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66245We consider an operator function (F(λ)) for (λ\in(σ,τ)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal N_F(α,β)) of the eigenvalues of (F(λ)) on a segment ([α,β)\Subset(σ,τ)). In particular, we prove the estimates (\mathcal N_F(α,β)\geqslant ν_F(β)-ν_F(α)) and (\mathcal N_F(α,β)= ν_F(β)-ν_F(α)) where (ν(ξ)) is the number of the negative eigenvalues of the operator (F(ξ)), (ξ\in(σ,τ)). The obtained results are applied for the functions of ordinary differential operators on a finite interval.9 pagesFunctional Analysis47B25; 34L15text