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

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 C0. 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 Carath´eodory 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.