Skew exactness perturbation

Robin Harte and David Larson The first author was partially supported by Enterprise Ireland grant number IC/2001/027 Abstract We offer a perturbation theory for finite ascent and descent properties of bounded operators. There are various degrees of “skew exactness” ([10];[7] (10.9.0.1), (10.9.0.2)) between compatible pairs of operators, bounded and linear between normed spaces: 1. Definition Suppose T : X → Y and S : Y → Z are bounded and linear between normed spaces; then we may classify the pair (S, T ) as left skew exact if there is inclusion 1.1 S−1(0) ∩ T (X) = {0} , strongly left skew exact if there is k > 0 for which 1.2 ‖T (·)‖ ≤ k‖ST (·)‖ , and splitting left skew exact if there is R ∈ BL(Z, Y ) for which 1.3 T = RST . Also we may classify the pair (S, T ) as right skew exact if there is inclusion 1.4 S−1(0) + T (X) = Y , strongly right skew exact if there is k > 0 for which: for every y ∈ Y there is x ∈ X for which 1.5 Sy = STx with ‖x‖ ≤ k‖y‖ , and splitting right skew exact if there is R ∈ BL(Y,X) for which 1.6 S = STR . It is easy to see that 2. Theorem In the notation of Definition 1, there is implication 2.1 (1.3) =⇒ (1.2) =⇒ (1.1)