Open Access

# Conservative L-systems and the Livšic function

### Abstract

We study the connection between the classes of (i) Livsic functions $s(z),$ i.e., the characteristic functions of densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$; (ii) the characteristic functions $S(z)$ of a maximal dissipative extension $T$ of $\dot A,$ i.e., the Mobius transform of $s(z)$ determined by the von Neumann parameter $\kappa$ of the extension relative to an appropriate basis in the deficiency subspaces; and (iii) the transfer functions $W_\Theta(z)$ of a conservative L-system $\Theta$ with the main operator $T$. It is shown that under a natural hypothesis {the functions $S(z)$ and $W_\Theta(z)$ are reciprocal to each other. In particular, $W_\Theta(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$ whenever $\kappa=0$. 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 ($\kappa=0$). 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.

Key words: L-system, transfer function, impedance function, Herglotz-Nevanlinna function, Weyl-Titchmarsh function, Livsic function, characteristic function, Donoghue class, symmetric operator, dissipative extension, von Neumann parameter.

### Article Information

 Title Conservative L-systems and the Livšic function Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 2, 104-133 MathSciNet 3407905 zbMATH 06533471 Milestones Received 27/01/2015; Revised 13/02/2015 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

S. Belyi
Department of Mathematics, Troy State University, Troy, AL 36082, USA

K. A. Makarov
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E. Tsekanovskii
Department of Mathematics Niagara University, NY 14109, USA

### Citation Example

S. Belyi, K. A. Makarov, and E. Tsekanovskiĭ, Conservative L-systems and the Livšic function, Methods Funct. Anal. Topology 21 (2015), no. 2, 104-133.

### BibTex

@article {MFAT773,
AUTHOR = {Belyi, S. and Makarov, K. A. and Tsekanovskiĭ, E.},
TITLE = {Conservative L-systems and the Livšic function},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {2},
PAGES = {104-133},
ISSN = {1029-3531},
MRNUMBER = {3407905},
ZBLNUMBER = {06533471},
URL = {http://mfat.imath.kiev.ua/article/?id=773},
}

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