Linear transformation of Tauberian type in normed spaces


Let T : D(T) ⊂ X → Y be a linear transformation where X and Y are normed spaces. We call T Tauberian if (T'')<sup>-1</sup>(Q\hat{Y})⊂ \tilde{D}(T)<sup>\wedge</sup> where Q is the quotient map defined on Y'' with kernel D(T')<sup>\bot</sup>.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<sub>p</sub>, 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.

DOI Code: 10.1285/i15900932v10supn1p193

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