2014/06/10 by S. Belyi, Belyi, S., K. A. Makarov +3
Mathematics · #47N50 #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1406.2399
openalex publication_date 2014/06/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We study the connection between the Livšic class of functions s(z) that are the characteristic functions of densely defined symmetric operators A with deficiency indices (1, 1), the characteristic functions S(z) (the Möbius transform of s(z)) of a maximal dissipative extension T of A (determined by the von Neumann parameter κ of the extension relative to an appropriate basis in the deficiency subspaces) and the transfer functions WΘ(z) of a conservative L-system Θ with the main operator T. It is shown that under a natural hypothesis S(z) and WΘ(z) are reciprocal to each other. In particular, when κ=0, WΘ(z)=(1)/(S(z))=-(1)/(s(z)). It is established that the impedance function of a conservative L-system with the main operator T coincides with the function from 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. All results are illustrated by a number of examples.