  A semiregular operator on a Hilbert C^*-module, or equivalently, on the
C^*-algebra of `compact' operators on it, is a closable densely defined
operator whose adjoint is also densely defined. It is shown that for operators
on extensions of compacts by unital or abelian C^*-algebras, semiregularity
leads to regularity. Two examples coming from quantum groups are discussed.
