Dissipative operator and its Cayley transform


Abstract: In this paper, we investigate the spectral properties of the maximaldissipative extension of the minimal symmetric differential operatorgenerated by a second order differential expression and dissipative andeigenparameter dependent boundary conditions. For this purpose we use thecharacteristic function of the maximal dissipative operator and inverseoperator. This investigation is done by the characteristic function of theCayley transform of the maximal dissipative operator, which is a completelynonunitary contraction belonging to the class $C_{0}.$ Using Solomyak'smethod we also introduce the self-adjoint dilation of the maximal dissipativeoperator and incoming/outgoing eigenfunctions of the dilation. Moreover, weinvestigate other properties of the Cayley transform of the maximaldissipative operator.

Keywords: Cayley transform, completely nonunitary contraction, unitary colligation, characteristic function, CMV matrix

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