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.

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Copyright © 2013 Elsevier Inc. This article first appeared in Journal of Functional Analysis 264:4 (2013), 999-1048.

The definitive version is available at:

DOI: 10.1016/j.jfa.2012.11.015

Full Citation:

Aleman, Alexandru, R.T.W. Martin, and William T. Ross. "On a Theorem of Livsic." Journal of Functional Analysis 264, no. 4 (2013): 999-1048. doi:10.1016/j.jfa.2012.11.015.