On a theorem of Livsic

The theory of symmetric operators has several deep applications to the function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions (model subspaces of Hardy spaces, de Branges–Rovnyak spaces, Herglotz spaces), Sturm–Liouville and Schrodinger differential operators, 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. Moreover, we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator.