Regular operators on Hilbert C*-modules

A regular operator T on a Hilbert C*-module is defined just like a closed o perator 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 operat ors 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 onesel f 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 land restrictions) are related. Finally, using these results, we give a criterion for a semiregular operato r on a liminal C*-algebra to have a regular extension.