Normal Differential Operators of Third Order

Abstract

In the Hilbert space of vector-functions L2(H,(a, b)), where H is anyseparable Hilbert space, the general representation in terms of boundary values of all normal extensions of the formally normal minimaloperator, generated by linear differential-operator expressions of thirdorder in the forml(u) = u′′′(t) + A3u(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, thecase of A−1 ∈ S∞(H), asymptotic estimates of normal extensions ofeigenvalues has been established at infinity.