A structure theorem for echelon Köthe spaces
Abstract
In [9], Lopez-Molina defined the echelon Köthe spaces
, which provide a suitable generalization of the echelon sequence spaces
to a general measure space
(see also [5]). In this paper, we show that the strutture of the separable echelon Köthe spaces is «nicely» close to the strutture of the echelon sequence spaces. Namely, our main result is: Let
be a separable echelon Köthe space of order p,
, with
purely non-atomic. Then, there is an echelon sequence space
so that
is isomorphic to the space
of all
-summable sequences in
. An an application we show that a separable echelon Köthe space has a basis (by a basis we mean a Schauder basis), which is unconditional if
.
![Λ<sup>p</sup>(\Chi,𝛽,\mu,g<sub>k</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4c076332fee433a190f3514c0f637746.png)
![λ<sup>p</sup>(a_{n}<sup>k</sup>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/abf3a36c1b591f6ccb1c719be7fc0ad2.png)
![\Chi, 𝛽, \mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/3bec31ef08bda914108f48b1592af82e.png)
![Λ<sup>p</sup>(\Chi,𝛽,\mu,g<sub>k</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4c076332fee433a190f3514c0f637746.png)
![1 ≤ p ≤ ∈fty](http://siba-ese.unile.it/plugins/generic/latexRender/cache/3842bcc4d66cdb6a14af44c1efba27a1.png)
![(\Chi,𝛽,\mu,g<sub>k</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6395c70adb7f48c23f3ef28859e1593f.png)
![λ<sup>p</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/aa77a97772a3a2e41e77da2e000bc331.png)
![Λ<sup>p</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/bebbb69a637fc76e8f40698ddaab8bb1.png)
![λ<sup>p</sup>(L<sup>p</sup>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c571a93538df8a3225b7a843400d6ba3.png)
![λ<sup>p</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/aa77a97772a3a2e41e77da2e000bc331.png)
![L<sup>p</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/7b9dabe4a872a8aaf123760be1ed9a11.png)
![p > 1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c30a4409f92a86906dae6d31e51c5408.png)
DOI Code:
10.1285/i15900932v9n2p165
Full Text: PDF