Conservative L-systems and the Livšic function - Makarov, K. A.; Belyi, S.; Tsekanovskii, E. (Methods of Functional Analysis and Topology, 2015)
We study the connection between the classes of (i) Livsic functions s(z), i.e., the characteristic functions of densely defined symmetric operators A with deficiency indices (1, 1); (ii) the characteristic functions S(z) of a maximal dissipative extension T of A, i.e., the Mobius transform of s(z) determined by the von Neumann parameter κ of the extension relative to an appropriate basis in the deficiency subspaces; and (iii) the transfer functions W_Θ(z) of a conservative L-system Θ with the main operator T. It is shown that under a natural hypothesis the functions S(z) and W_Θ(z) are reciprocal to each other. In particular, W_Θ(z)=1S(z)=-1s(z) whenever κ=0. It is established that the impedance function of a conservative L-system with the main operator T belongs to the Donoghue class if and only if the von Neumann parameter vanishes (κ=0). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. We also obtain the representation of a function from this class via the Weyl-Titchmarsh function. All results are illustrated by a number of examples.