Abstract: Let H₁ and H₂ be Hilbert spaces. In this paper, the authors present some equivalent conditions for the existence of generalized inverse A⁽²⁾_T,S of a bounded linear operator A ∈ B(H₁,H₂), which is a bounded linear operator X ∈ B(H₂,H₁) satisfying
XAX=X, R(X)=T, N(X)=S.
The basic properties and some expressions of the generalized inverse A⁽²⁾_T,S are also investigated.

Key words: Hilbert space, Linear operator, Generalized inverses A⁽²⁾_T,S