The L-system representation and c-entropy

TL;DR

This work establishes a constructive correspondence between Weyl-Titchmarsh functions of a symmetric operator with deficiency indices $(1,1)$ and impedance functions of uniquely determined L-systems with main operators in the same Hilbert space. It provides explicit realizations for $M_{({\dot A},A)}(z)$ and $M_{({\dot A},A_\alpha)}(z)$, and extends to the renormalized $aM_{({\dot A},A)}(z)$, detailing how the coupling parameter $\kappa$ and the quasi-kernel parameter $U$ encode these realizations. A central contribution is the analysis of c-entropy and the dissipation coefficient, including exact formulae for $\mathcal{S}(a)$ and $\mathcal{D}(a)$, their invariance properties, and their interpretation via Abelian differentials. The paper culminates in concrete examples that illustrate the construction and demonstrate cases of infinite and finite entropy depending on $a$, strengthening the link between operator theory, system theory, and complex function theory in the Donoghue framework.

Abstract

Given a symmetric operator $\dot A$ with deficiency indices $(1,1)$ and its self-adjoint extension $A$ in a Hilbert space $\mathcal{H}$, we construct a (unique) L-system with the main operator in $\mathcal{H}$ such that its impedance mapping coincides with the Weyl-Titchmarsh function $M_{(\dot A, A)}(z)$ or its linear-fractional transformation $M_{(\dot A, A_α)}(z)$. Similar L-system constructions are provided for the Weyl-Titchmarsh function $aM_{(\dot A, A)}(z)$ with $a>0$. We also evaluate c-entropy and the main operator dissipation coefficient for the obtained L-systems.

Theorems & Definitions (23)

• Definition 1
• Proposition 2: BMkTBMkT-2
• Remark 3
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7
• proof
• Theorem 8