Linear transformation of Tauberian type in normed spaces

Abstract

Let $T : D(T) ⊂ X → Y$ be a linear transformation where X and Y are normed spaces. We call T Tauberian if $(T'')^-1(Q\hat{Y})⊂ \tilde{D}(T)^\wedge$ where Q is the quotient map defined on $Y''$ with kernel $D(T')^\bot$.Bounded Tauberian operators in Banach spaces were studied by Kalton and Wilansky in [KW]. As Gonzalez and Onieva remark in [G03], these operators appear in summability (see [GW]), factorization of operators [DFJP], [N], preservation of isomorphic properties of Banach spaces [N], the preservation of the closed ness of images of closed sets [NR], the equivalence between the Radon-Nikodym property and the Krein-Milman property [S], and generalized Fredholm operators [T], [Y].Classes of Tauberian operators related to a certain measure of weak compactness are investigated in [AT].Other recent works are [AG] (which contains the solution of a problem raised in [KW]), [Gon1], [Gon2], [GO1], [G02], [G03], and [MP].The present paper investigates unbounded Tauberian operators.This wider class is a natural object of study in any investigation concerning the second adjoint $T''$ of an unbounded operator, about which little seems to be known.Our main goal is Theorem 3.10 which implies as a corollary the following partial characterization: Let $T'$ be continuous. Then T is Tauberian if and only if for each bounded subset B of $D(T)$, if $TB$ is relatively $σ(Y, D( T'))$ compact (alternatively, relatively $D( T')$ -seminorm compact) then B is relatively $σ (\tilde {D}(T), D(T)')$ compact.This result contains the well known characterization [KW; Theorem 3.2] for the classical case. Section 4 provides some examples and further properties of Tauberian operators; thus for example the usual closable ordinary differential operators defined between $L_p$, spaces (see e.g. [Go1; Ch VI]) and their successive adjoints are all Tauberian (Corollaries 4.6 and 4.7). Section 5 looks at the continuous case.