  We develop a duality theory for unbounded Hermitian operators with dense
domain in Hilbert space. As is known, the obstruction for a Hermitian operator
to be selfadjoint or to have selfadjoint extensions is measured by a pair of
deficiency indices, and associated deficiency spaces; but in practical
problems, the direct computation of these indices can be difficult. Instead, in
this paper we identify additional structures that throw light on the problem.
While duality considerations are a tested tool in mathematics, we will attack
the problem of computing deficiency spaces for a single Hermitian operator with
dense domain in a Hilbert space which occurs in a duality relation with a
second Hermitian operator, often in the same Hilbert space.
