2020/06/29 by Rytis Juršėnas, Jursenas, R.
Mathematics · #46C20 #47A06 #47B25 #47B50 #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2006.15964
openalex publication_date 2020/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (\mathfrakL,Γ) be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space \mathfrakH. Let MΓ be the Weyl family corresponding to (\mathfrakL,Γ). We cope with two main topics. First, since MΓ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation MΓ(z), for some z∈ℂ\smallsetminusℝ, becomes a nontrivial task. Regarding MΓ(z) as the (Shmul'yan) transform of zI induced by Γ, we give conditions for the equality in MΓ(z)⊆MΓ(z) to hold and we compute the adjoint MΓ(z)^*. As an application we ask when the resolvent set of the main transform associated with a unitary boundary pair for T+ is nonempty. Based on the criterion for the closeness of MΓ(z) we give a sufficient condition for the answer. It follows, for example, that, if T is a standard linear relation in a Pontryagin space then the Weyl family MΓ corresponding to a boundary relation Γ for T+ is a generalized Nevanlinna family. In the second topic we characterize the transformed boundary pair (\mathfrakL^′,Γ^′) with its Weyl family MΓ^′. The transformation scheme is either Γ^′=ΓV-1 or Γ^′=VΓ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between (\mathfrakL,Γ) and (\mathfrakL^′,Γ^′); the formula for MΓ^′-MΓ, for an ordinary boundary triple and a standard unitary operator V; construction of a quasi boundary triple from an isometric boundary triple (\mathfrakL,Γ0,Γ1) with kerΓ=T and T0=T^*0.