Linear transformation of Tauberian type in normed spaces
Abstract
Let
be a linear transformation where X and Y are normed spaces. We call T Tauberian if
where Q is the quotient map defined on
with kernel
.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
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
be continuous. Then T is Tauberian if and only if for each bounded subset B of
, if
is relatively
compact (alternatively, relatively
-seminorm compact) then B is relatively
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
, 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.
![T : D(T) ⊂ X → Y](http://212.189.136.205/plugins/generic/latexRender/cache/369e535145483812ededefe0dc6745ed.png)
![(T'')<sup>-1</sup>(Q\hat{Y})⊂ \tilde{D}(T)<sup>\wedge</sup>](http://212.189.136.205/plugins/generic/latexRender/cache/d3a78e45fe2d8b9ee1391ea7516ebdcc.png)
![Y''](http://212.189.136.205/plugins/generic/latexRender/cache/318d8cc6e88274b594811e0bae41b91a.png)
![D(T')<sup>\bot</sup>](http://212.189.136.205/plugins/generic/latexRender/cache/b7a257db76fe5d554810416d27b0a6ff.png)
![T''](http://212.189.136.205/plugins/generic/latexRender/cache/422bb1ecfea58b6539aa8fd638047cf9.png)
![T'](http://212.189.136.205/plugins/generic/latexRender/cache/6f3357ae1d6de5c7df30cf8503177f87.png)
![D(T)](http://212.189.136.205/plugins/generic/latexRender/cache/1f27565bb2699ada2c7c6cc7c81d1436.png)
![TB](http://212.189.136.205/plugins/generic/latexRender/cache/ff88442a425c06d961f97bccb11ddf5d.png)
![σ(Y, D( T'))](http://212.189.136.205/plugins/generic/latexRender/cache/7a9167f481292fc1bbeb93bab94ff039.png)
![D( T')](http://212.189.136.205/plugins/generic/latexRender/cache/9be85dccd1264f8bec36b07ca454de3f.png)
![σ (\tilde {D}(T), D(T)')](http://212.189.136.205/plugins/generic/latexRender/cache/2ec9a176e1ed4fd09a3c489540a5721a.png)
![L<sub>p</sub>](http://212.189.136.205/plugins/generic/latexRender/cache/231e9296852667df4a7d7d6f3dc99368.png)
DOI Code:
10.1285/i15900932v10supn1p193
Full Text: PDF