A structure theorem for echelon Köthe spaces


In [9], Lopez-Molina defined the echelon Köthe spaces Λ<sup>p</sup>(\Chi,𝛽,\mu,g<sub>k</sub>), which provide a suitable generalization of the echelon sequence spaces λ<sup>p</sup>(a_{n}<sup>k</sup>) to a general measure space \Chi, 𝛽, \mu (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 Λ<sup>p</sup>(\Chi,𝛽,\mu,g<sub>k</sub>) be a separable echelon Köthe space of order p, 1 ≤ p ≤ ∈fty, with (\Chi,𝛽,\mu,g<sub>k</sub>) purely non-atomic. Then, there is an echelon sequence space λ<sup>p</sup> so that Λ<sup>p</sup> is isomorphic to the space λ<sup>p</sup>(L<sup>p</sup>) of all λ<sup>p</sup>-summable sequences in L<sup>p</sup>. 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 p > 1.

DOI Code: 10.1285/i15900932v9n2p165

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