On the asymptotic expression of the number of eigenvalues of differential equation with the operator

In this study we prove the pure discrete property of the spectrum and found the asymptotic expression of N (λ) the number of eigenvalues < λ ( λ > 0 ) when λ → ∞ of Loperator in space L2(0,∞;H) . L operator is formed by + Σ − + < < ∞ = y P (x)y(3 j) Q(x)y , 0 x j2j 1 IV differential expression with boundary conditions y (0) by(0) 0y (0) ay (0) 0′′′ − =′′ − ′ = . Here H is a separable Hilbert space, a,b are real constants, the operator valued functions Q(x) , Pj(x) (j = 1,2) are defined in the H Hilbert space and satisfy the following conditions Q* (x) = Q(x) ≥ I ( I is the unit operator in H) , ∞ Q−1(x) ∈ σ , P (x)Q 4 (x) c (j 1,2 c const. 0; 0).