Norm-attainment of linear functionals on subspaces and characterizations of Tauberian operators

Abstract

Introduction. Let X and Y be Banach spaces and T: X -> 7 a given non-zero operator. (Throughout, "operator" means "bounded linear map".) Under what circumstances is it true that TK is closed for every closed bounded convex subset K of XΊ Evidently this is trivially true if X is reflexive, so suppose this is not the case. Here are some of the equivalences obtained in our main result of §2, Theorem 2.3. (For any Banach space Z, let B z = {z e Z: ||z|| < 1}.)

Characterizations of Tauberian operators and other semigroups of operators

“…Also SWCo+ coincides with the class of all Tauberian operators [9,12], and SCC+ with the class of all operators preserving mere Cauchy sequences considered in [10]. Moreover for T in SU+ (SU_) we have N(T) e Sp{U) (Y/RJJ) e Sp{Ud)), and T + K is in SU+ (SU_) for every operator K in U (U ).…”

“…Recently Neidinger and Rosenthal [12] have obtained characterizations of Tauberian operators in terms of the closedness of images of closed sets.…”

Cotauberian Operators on L 1(0, 1) Obtained by Lifting

“…Tauberian operators were introduced in [13] as those operators T : X → Y such that the second conjugate satisfies T * * −1 (Y ) = X. They have found many applications in Banach space theory like factorization of operators [5], preservation of isomorphic properties [16], equivalence between the Radon-Nikodym property and the Krein-Milman property [18], and refinements of James' characterization of reflexive spaces [17]. The cotauberian operators were introduced by Tacon [19] as those operators T such that T * is tauberian, and they have found applications in factorization of operators and preservation of isomorphic properties of Banach spaces (see [8]).…”

Some examples of Tauberian operators

“…Recently they have received some attention because they form a broader class than that of isomorphisms (into), but yet they preserve some isomorphic properties of Banach spaces [8,9].…”