Abstract
In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential equation and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal nonselfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.
Highlights
A model operator may be regarded as an equivalent operator to another operator in a certain sense
Solomyak showed for a maximal dissipative operator A and its Cayley transform C(A) that there exist isometric isomorphisms ρ : F(A) → DC, ρ∗ : F∗(A) → DC∗
In this paper using the results of Solomyak we investigate some spectral properties of a regular third-order dissipative operator
Summary
A model operator may be regarded as an equivalent operator to another operator in a certain sense. It is known that a dissipative operator is maximal if and only if C(A) is a contraction such that domain of C(A) is the Hilbert space K and 1 can not belong to the point spectrum of C(A). Solomyak used these connections and boundary spaces associated with A to construct the characteristic function SA(λ) with the rule. Solomyak showed for a maximal dissipative operator A and its Cayley transform C(A) that there exist isometric isomorphisms ρ : F(A) → DC, ρ∗ : F∗(A) → DC∗. This paper may give an idea to use Solomyak’s method for the oddorder dissipative or accumulative operators
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