Characteristic function of M. S. Livšic and triangular models of bounded linear operators

TL;DR

The paper develops M. S. Livšic's characteristic function for non-selfadjoint operators through the framework of operator colligations and open systems, showing that the characteristic function is a complete unitary invariant and encodes invariant subspaces via left divisors. It introduces the $J$-property, classifies characteristic functions into $\Omega_J$ and its quasihermitian/finitary variants, and establishes factorization theorems that underlie triangular Livšic models. By linking multiplication of colligations to multiplication of transfer functions and employing multiplicative integrals, the work constructs triangular models and derives applications to dissipative operators, including a discrete, finite-dimensional representation. Overall, the results provide a robust, algebraic-analytic toolkit for spectral and model-theoretic analysis of non-selfadjoint operators with energy-interacting open systems.

Abstract

This paper is dedicated to the introduction in a circle of ideas and methods, which are connected with the notion of characteristic function of a non-selfadjoint operator. We start with the consideration of closed and open systems (Subsections 2.1.1-2.1.2). In Subsections 2.1.2-2.1.3 we introduce the notion of operator colligation and define the characteristic function of the operator colligation as transfer function of the corresponding open system. In Section 3 we state three basic properties of the c.o.f.. First (Subsection 3.1), we note that the c.o.f. is the full unitary invariant of the operator colligation. Second (see Theorem 3.4), it turns out that the invariant subspaces of the corresponding operator are associated with left divisors of the c.o.f.. Third, the $J$-property of the c.o.f. (see (3.6)-(3.8)) is a basic property which determines the class of c. o. f. (see Section 4). In Chapter 4 we describe the classes of characteristic functions which play an important role in our considerations. In Chapter 5 we state necessary facts on multiplicative integral. Chapter 6 is devoted to the factorization theorem (Theorem 6.7) for matrix-valued characteristic function. In Chapter 7 we construct a triangular Livšic model of bounded linear operator and as application we obtain some known results on dissipative operators.

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Theorems & Definitions (133)

• Definition 2.1: MR0347396, MR0634096
• Theorem 2.2: MR0322542
• proof
• Remark 2.3
• Definition 2.4: MR0062955
• Remark 2.5
• Definition 2.6
• Theorem 2.7: MR0322542
• proof
• Theorem 2.8: MR0182516