Locally adjointable operators on Hilbert C^*-modules

In the theory of Hilbert C^*-modules over a C^*-algebra A (in contrast with the theory of Hilbert spaces) not each bounded operator (A-homomorphism) admits an adjoint. The interplay between the sets of adjointable and non-adjointable operators plays a very important role in the theory. We study an intermediate notion of locally adjointable operator F: M N, i. e. such an operator that F g is adjointable for any adjointable g: A M. We have introduced this notion recently and it has demonstrated its usefulness in the context of theory of uniform structures on Hilbert C^*-modules. In the present paper we obtain an explicit description of locally adjointable operators in important cases.