QUOTIENTS OF ADJOINTABLE OPERATORS ON HILBERT C-MODULES

Let T and S be bounded adjointable operators on a Hilbert C*-module E such that ker(S) is orthogonally complemented in E. We prove that the quotient TS–1 is a closed operator with orthogonally complemented graph in E ⊕ E if and only if ran(T*) + ran(S*) is closed. We mean here by S–1 the inverse of the restriction of S to ker(S)⊥. This leads us to study the operators as TS†, whenever S admits the Moore–Penrose inverse S†. Note that in case of an injective Moore–Penrose invertible operator S, we have S–1 = S†. Then we present some applications of these results. Moreover, the quotients of regular operators are also investigated in this paper.