On the unitary equivalence of the proper extensions of a Hermitian operator and the Weyl function

Suppose that A is a symmetric densely defined operator in a Hilbert space H with equal deficiency indices. In the last 30 years, the approach based on the notions of boundary triplet of an operatorA∗ and of the corresponding Weyl function has become quite popular in extension theory (see [1]–[3]). It is well known (see [2]) that, in the case a simple operator A, the Weyl function M( · ) of the boundary triplet Π = {H ,Γ0,Γ1} uniquely determines this triplet up to unitary equivalence. In particular,M( · ) defines the pair {A,A0}, in which A0 := A∗ ker Γ0 = A0 is uniquely defined up to unitary similarity. On the other hand, if ΠD = {H0 ⊕H1,Γ,Γ } is the boundary triplet of the dual pair {A,A } of operators, then the corresponding Weyl function defines the triplet ΠD only up to weak similarity (see [4]). The conditions on the pair {A,A }, under which the Weyl function defines the triplet ΠD up to similarity were obtained in the recent papers [5]–[7].