tailieunhanh - Dissipative operator and its Cayley transform

In this paper, we investigate the spectral properties of the maximal dissipative extension of the minimal symmetric differential operator generated by a second order differential expression and dissipative and eigenparameter dependent boundary conditions. For this purpose we use the characteristic function of the maximal dissipative operator and inverse operator. | Turk J Math (2017) 41: 1404 – 1432 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Dissipative operator and its Cayley transform ∗ ˘ Ekin UGURLU , Kenan TAS ¸ Department of Mathematics, Faculty of Arts and Sciences, C ¸ ankaya University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we investigate the spectral properties of the maximal dissipative extension of the minimal symmetric differential operator generated by a second order differential expression and dissipative and eigenparameter dependent boundary conditions. For this purpose we use the characteristic function of the maximal dissipative operator and inverse operator. This investigation is done by the characteristic function of the Cayley transform of the maximal dissipative operator, which is a completely nonunitary contraction belonging to the class C0 . Using Solomyak’s method we also introduce the self-adjoint dilation of the maximal dissipative operator and incoming/outgoing eigenfunctions of the dilation. Moreover, we investigate other properties of the Cayley transform of the maximal dissipative operator. Key words: Cayley transform, completely nonunitary contraction, unitary colligation, characteristic function, CMV matrix 1. Introduction If an operator T1 acting on a Hilbert space H1 is equivalent to another operator T2 acting on another Hilbert space H2 in a certain sense, then one can say that T2 is a model of T1 . There exist models up to unitary equivalence, similarity equivalence, quasi-similarity, pseudo-similarity, and other equivalences [27]. A useful model was given by and Foia¸s [24, 25]. and Foia¸s constructed the model operator for the contractive operators acting on Hilbert spaces. This construction is based on the dilation. An operator U acting on a Hilbert space H is called a dilation of an .