Unitary equivalence of proper extensions of a symmetric operator and the Weyl function

Abstract

Let $A$ be a densely defined simple symmetric operator in $\gH$, let $Π=\bt$ be a boundary triplet for $A^{\ast }$ and let $M(\cd)$ be the corresponding Weyl function. It is known that the Weyl function $M(\cd)$ determines the boundary triplet $\Pi$, in particular, the pair $A,A_0$, where $A_0:= A^*\lceil\ker\G_0 (= A^*_0)$, uniquely up to unitary similarity. At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to weak similarity. In this paper we consider symmetric dual pairs $A,A$ generated by $A\subset A^{\ast }$ and special boundary triplets $\wtΠ$ for $A,A$. We are interested whether the result on unitary similarity remains valid provided that the Weyl function corresponding to $\wtΠ$ is $\wt M(z)= K^*(B-M(z))^{-1} K,$ where $B$ is some non-self-adjoint bounded operator in $\cH$. We specify some conditions in terms of the operators $A_0$ and $A_B= A^*\lceil \ker(\G_1-B\G_0)$, which determine uniquely (up to unitary equivalence) the pair $A,A_B$ by the Weyl function $\wt M(\cd)$. Moreover, it is shown that under some additional assumptions the Weyl function $M_{\Pi }(\cdot )$ of the boundary triplet $\Pi$ for the dual pair $\DA$ determines the triplet $\Pi$ uniquely up to unitary similarity. We obtain also some negative results demonstrating that in general the Weyl function $\wt M(\cd)$ does not determine the operator $A_B$ even up to similarity.