Conservative L-systems and the Livšic functions. (English) Zbl 1340.47023

The authors study the connection between the classes of (i) Livšic functions, that is characteristic functions of densely defined symmetric operators \(A\) with deficiency indices (1,1); (ii) characteristic functions of maximal dissipative extensions \(T\) of \(A\); (iii) the transfer function \(W_\Theta (z)\) of a conservative \(L\)-system (an operator colligation) with the main operator \(T\); see [Y. Arlinskii et al., “Conservative realizations of Herglotz-Nevanlinna functions”, Operator Theory: Advances and Applications 217. Basel: Birkhäuser (2011; Zbl 1240.47001)].
Analytic properties of the impedance function \(V_\Theta (z)=i[W_\theta (z)+I]^{-1}[W_\theta (z)-I]\) are studied, in particular, its belonging to the Donoghue class or its generalizations. Functions from the generalized Donoghue class are also represented via the Weyl-Titchmarsh functions associated with \(A\) and its extensions.


47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
47B25 Linear symmetric and selfadjoint operators (unbounded)
93B28 Operator-theoretic methods


Zbl 1240.47001
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