Adjoint pairs and unbounded normal operators

Abstract An adjoint pair is a of densely defined closed linear operators A,B on Hilbert space such that $$ \langle Ax,y\rangle = x,By\rangle$$ ⟨ A x , y ⟩ = B for $$x \in \mathcal{D} (A),y (B) ∈ D ( ) . We consider pairs which 0 regular point both and associate boundary triplet to an pair. Proper extensions the operator B are in one-to-one correspondence $$Tc\leftrightarrow \mathcal{C} T c ↔ C subspaces $$\mathcal{C} \mathcal{N}(A^{*}) \oplus\mathcal{N}(B^{*})$$ N ∗ ⊕ In case when formally normal $$\mathcal{D}(A) =\mathcal{D}(B) , $$Tc Tc characterized. Next we assume has extension with bounded inverse. Then $$Tc$$ described $$\mathcal{N}(A^{*}) dimension one treated.