Seminorms related to weak compactness and to Tauberian operators

Abstract

Let E and F be Banach spaces. The semigroup Φ + ( E , F ) of semi-Fredholm operators consists of the bounded linear mappings E → F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T ∈Φ + ( E , F ) if and only if for any bounded set B ⊂ E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ + ( E , F ) if and only if there is c ≥ 0 such that for all bounded B ⊂ E . Here γ is the well known Hausdorff measure of non-compactness with B E the closed unit ball of E .