2012/08/06 by Seppo Hassi, M. M. Malamud, Hassi, Seppo +3 · 1 citation
Mathematics · #47A56 #47B25 (Primary) 47A48 #47E05 (Secondary) #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1208.1201
openalex publication_date 2012/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a densely defined simple symmetric operator in \gH, let Π=\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 Π, in particular, the pair A,A0, where A0:= A^*\lceilker\G0 (= 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⊂ A^* 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 A0 and AB= A^*\lceil ker(\G1-B\G0), which determine uniquely (up to unitary equivalence) the pair A,AB by the Weyl function \wt M(\cd). Moreover, it is shown that under some additional assumptions the Weyl function MΠ(⋅) of the boundary triplet Π for the dual pair \DA determines the triplet Π 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 AB even up to similarity.