On a completely non-unitary contraction and associated dissipative difference operator

Abstract

In this paper, we investigate the spectral properties of dissipative difference operator, dissipative sum operator and contractive operator. Using Solomyak's method, we construct the characteristic function of the dissipative difference operator. For this purpose, we use boundary spaces and functional embeddings. Then we pass to the characteristic function of the Cayley transform of the dissipative difference operator which is a completely non-unitary contraction belonging to the class C 0 . With the aid of this characteristic function we achieve to pass to the minimal function of the contraction and we investigate the complete spectral analysis of both the contractive and dissipative operators. Embedding the associated contraction to its natural unitary colligation, we obtain a Carathéodory function. Moreover, self-adjoint dilation of the maximal dissipative difference operator and its incoming and outgoing eigenfunctions are constructed. Finally, the truncated CMV matrix is established which is unitary equivalent to the contractive operator.

The spectral analysis of a system of first‐order equations with dissipative boundary conditions

“…In this paper, we will consider a dissipative operator related with a non‐self‐adjoint boundary value problem and to get some information about spectral properties of the operator we will construct the characteristic function of the dissipative operator using Solomyak's method 11 . This method has been applied to second‐order difference operator in Uğurlu and Baleanu, 12 third‐order differential operator in Uğurlu and Baleanu, 13 and fourth‐order differential operator in Uğurlu 14 . We shall also note that the readers may find some papers containing the spectral analysis of some dissipative operators in, for example 15–20 .…”

Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator

“…We should note that such an investigation with the aid of Solomyak's approach has not been introduced for the thirdorder case. In fact, the literature has less works on odd-order operators than on even-order equations even if there exists some results in the literature [6][7][8][9][10][11][12][13]. The main reason is the confusion of imposing the boundary conditions because as Everitt says in [9] that it is impossible to impose separated boundary conditions for the solutions of a third-order equation.…”