DOUGLAS RANGE FACTORIZATION THEOREM FOR REGULAR OPERATORS ON HILBERT C-MODULES

In this paper, we aim to extend the Douglas range factorization theorem from the context of Hilbert spaces to the context of regular operators on a Hilbert C*-module. In particular, we show that if t and s are regular operators on a Hilbert C*-module E such that ran(t) ⊆ ran(s) and if s has a generalized inverse s✝, then r = s✝t is a densely defined operator satisfying t = sr. Moreover, if s is boundedly adjointable, then r is closed densely defined and its graph is orthogonally complemented in E ⊕ E, and if t is boundedly adjointable, then r is boundedly adjointable.