Adjoint pairs and unbounded normal operators

Abstract

An adjoint pair is a pair of densely defined closed linear operatorsA,B on a Hilbert space such that langle Ax,yrangle = langle x,Byrangle for x in mathcal{D} (A),y in mathcal{D} (B) . We consider adjoint pairs for which 0 is a regular point for bothoperators and associate a boundary triplet to such an adjoint pair. Properextensions of the operator B are in one-to-one correspondence Tcleftrightarrow mathcal{C} to closed subspaces mathcal{C} of mathcal{N}(A^{*}) oplusmathcal{N}(B^{*}). In the case when B is formallynormal andmathcal{D}(A) =mathcal{D}(B) , the normal operators Tc are characterized.Next we assume that B has an extension to a normal operator withbounded inverse. Then the normal operators Tc are described and thecase when mathcal{N}(A^{*}) has dimension one is treated.