On the Weyl-Titchmarsh and Livšic functions

2013/01/19 by Makarov, Konstantin, Tsekanovskii, Eduard
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1301.4610

Abstract

We establish a mutual relationship between main analytic objects for the dissipative extension theory of a symmetric operator A with deficiency indices (1,1). In particular, we introduce the Weyl-Titchmarsh function \cM of a maximal dissipative extension A of the symmetric operator A. Given a reference self-adjoint extension A of A, we introduce a von Neumann parameter κ, |κ|<1, characterizing the domain of the dissipative extension A against \Dom (A) and show that the pair (κ, \cM) is a complete unitary invariant of the triple ( A, A, A), unless κ=0. As a by-product of our considerations we obtain a relevant functional model for a dissipative operator and get an analog of the formula of Krein for its resolvent.

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