Realization of Herglotz–Nevanlinna Functions by Conservative Systems

Abstract

This chapter deals with the realization theory of different classes of Herglotz– Nevanlinna operator-valued functions as impedance functions of linear conservative L-systems. Nowadays realizations of various classes of operator-valued functions play an important role in modern spectral and system theories. An overview of comprehensive analysis of the abovementioned L-systems with, generally speaking, unbounded operators that satisfy the metric conservation law is provided. The treatment of realization problems for Herglotz– Nevanlinna functions and their various subclasses when members of these Y. Arlinskii Department of Mathematics, East Ukrainian National University, Lugansk, Ukraine e-mail: yma@snu.edu.ua S. Belyi ( ) Department of Mathematics, Troy University, Troy, AL, USA e-mail: sbelyi@troy.edu E. Tsekanovskii Department of Mathematics, Niagara University, New York, NY, USA e-mail: tsekanov@niagara.edu © Springer Basel 2015 D. Alpay (ed.), Operator Theory, DOI 10.1007/978-3-0348-0667-1_50 779 780 Y. Arlinskii et al. subclass are realized as impedance functions of L-systems is presented. In particular, the conservative realizations of Stieltjes, inverse Stieltjes, and general Herglotz–Nevanlinna functions and their connections to L-systems of different types with accretive, sectorial, and accumulative state-space operators are considered. The detailed study of the subject is based upon a new method involving extension theory of linear operators with the exit into rigged Hilbert spaces. A one-to-one correspondence between the impedance of L-systems and related extensions of unbounded operators with the exit into rigged Hilbert spaces is established. This material can be of interest to researchers in the field of operator theory, spectral analysis of differential operators, and system theory. Introduction Consider the following system of equations 8 ˆ ˆ< ˆ ˆ: i d dt C A .t/ D KJ .t/; .0/ D x 2 H; C D 2iK .t/; (31.1) where A is a bounded linear operator from a Hilbert space H into itself, K is a bounded linear operator from a Hilbert space E (dim E < 1) into H, J D J D J 1 maps E into itself, ImA D KJK . If for a given continuous in E function .t/ 2 L2Œ0; 0 .E/ the functions .t/ 2 H and C.t/ 2 L2Œ0; 0 .E/ satisfy the system (31.1), then the following metric conservation law holds 2k . /k2 2k .0/k2 D Z