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

TL;DR

This paper investigates when the Weyl function associated with boundary triplets of symmetric operators uniquely determines the operators and boundary conditions, revealing conditions for unitary equivalence and limitations of this determination.

Contribution

It extends the understanding of Weyl functions by identifying conditions under which they uniquely determine symmetric operator extensions up to unitary equivalence.

Findings

Weyl function determines boundary triplet up to unitary similarity under certain conditions.

The Weyl function may not always determine the operator extension up to similarity.

Additional assumptions can ensure the Weyl function uniquely determines the boundary triplet.

Abstract

Let $A$ be a densely defined simple symmetric operator in $\gH$, let $\Pi=\bt$ be a boundary triplet for $A^*$ 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 $A\subset A^*$ and special boundary triplets $\wt\Pi$ for ${A,A}$. We are interested whether the result on unitary similarity remains valid provided that the Weyl function corresponding to $\wt\Pi$ 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.