Normal Differential Operators of Third Order

Abstract

In the Hilbert space of vector-functions L
2
(H,(a, b)), where H is any
separable Hilbert space, the general representation in terms of boundary values of all normal extensions of the formally normal minimal
operator, generated by linear differential-operator expressions of third
order in the form
l(u) = u
′′′(t) + A
3
u(t), A : D(A) ⊂ H → H, A = A
∗ ≥ E,
is obtained in the first part of this study. Then, some spectral properties of these normal extensions are investigated. In particular, the
case of A
−1 ∈ S∞(H), asymptotic estimates of normal extensions of
eigenvalues has been established at infinity.

Keywords

• Normal extension,Compact operator,Eigenvalue,Asymptotical behavior of eigenvalues,2000 AMS Classification: 47 A 20

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