Abstract
An adjoint pair is a pair of densely defined closed linear operators A,B on a Hilbert space such that \( \langle Ax,y\rangle = \langle x,By\rangle\) for \(x \in \mathcal{D} (A),y \in \mathcal{D} (B) \). We consider adjoint pairs for which 0 is a regular point for both operators and associate a boundary triplet to such an adjoint pair. Proper extensions of the operator B are in one-to-one correspondence \(Tc\leftrightarrow \mathcal{C} \) to closed subspaces \(\mathcal{C} \) of \( \mathcal{N}(A^{*}) \oplus\mathcal{N}(B^{*})\). In the case when B is formally normal and\(\mathcal{D}(A) =\mathcal{D}(B) \), the normal operators \(Tc \) are characterized. Next we assume that B has an extension to a normal operator with bounded inverse. Then the normal operators \(Tc\) are described and the case when \(\mathcal{N}(A^{*}) \) has dimension one is treated.
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Schmüdgen, K. Adjoint pairs and unbounded normal operators. Acta Sci. Math. (Szeged) 88, 449–467 (2022). https://doi.org/10.1007/s44146-022-00024-z
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DOI: https://doi.org/10.1007/s44146-022-00024-z