Ulm’s method

Ulm’s method is motivated by the following

Proposition 2.1. Let A ∈ L(X,Y) and B ∈ L(Y,X). The following statements are equivalent:

(i) ‖I−BA‖ < 1 ; (2.3) (ii) C := (BA)−1B is a left-inverse of A and

‖C‖ ≤ ‖B‖ 1− ‖I−BA‖ ;

(iii) D := A(BA)−1 is a right-inverse of B and

‖D‖ ≤ ‖A‖ 1− ‖I−BA‖ ;

(iv) the null space of A is zero: N (A) = {0} ;

(v) the range of B is X : R(B) = X . If (i) is true, then 1◦ A is also right-invertible if and only if its range coincide with Y: R(A) = Y.