On a theorem of Livsic

TL;DR

This paper develops a comprehensive representation theory for simple symmetric operators with equal deficiency indices, refining classical results and providing new formulas for their characteristic functions, with applications in complex analysis and mathematical physics.

Contribution

It extends and refines Livsic's classical theorems by offering an alternative proof and a new, more computable formula for the Livsic characteristic function.

Findings

Provides a new proof of Livsic's theorem on unitary equivalence

Introduces a more accessible formula for the Livsic characteristic function

Enhances understanding of symmetric operators in complex function theory

Abstract

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operators, and infinite Jacobi matrices. In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and obtain a collection of results which refine and extend classical works of Krein and Livsic. In particular we provide an alternative proof of a theorem of Livsic which characterizes when two simple symmetric operators with equal deficiency indices are unitarily equivalent, and we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator with equal deficiency indices.