  A regular operator T on a Hilbert C^*-module is defined just like a closed
operator on a Hilbert space, with the extra condition that the range of
(I+T^*T) is dense. Semiregular operators are a slightly larger class of
operators that may not have this property. It is shown that, like in the case
of regular operators, one can, without any loss in generality, restrict oneself
to semiregular operators on C^*-algebras. We then prove that for abelian
C^*-algebras as well as for subalgebras of the algebra of compact operators,
any closed semiregular operator is automatically regular. We also determine how
a regular operator and its extensions (and restrictions) are related. Finally,
using these results, we give a criterion for a semiregular operator on a
liminal C^*-algebra to have a regular extension.
