Rank theorems of operators between Banach spaces

Let E and F be Banach spaces, and B(E, F) all of bounded linear operators on E into F. Let T(0)is an element of B(E, F) with an outer inverse T-0(#) is an element of B( F, E). Then a characteristic condition of S =(I + T-0(#)(T - T-0))(-1) T-0(#) with T is an element of B(E, F) and parallel to T-0(#) (T- T-0) parallel to < 1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.