The Canonical Spectral Measure in Köthe Echelon Spaces

Bonet, José; Ricker, Werner J.

doi:10.1007/s00020-005-1360-x

Cited by 9 publications

(8 citation statements)

References 15 publications

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“…Peris proved that the Fréchet spaces ℓ p+ , for 1 ≤ p < ∞, are not (FBa) spaces, thus providing a natural and concrete class of spaces of this type, [19]. For every 1 ≤ p < ∞, the Fréchet space ℓ p+ has the property that every ℓ p+ -valued vector measure (always assumed to be countably additive and dened on a σ-algebra) has relatively compact range if and only if p ∈ [1,2), [7,Proposition 2.8]. Once again the optimal solid extension ces(p+) of ℓ p+ exhibits better behaviour in this regard.…”

Section: Further Properties Of Ces(p+)mentioning

confidence: 99%

The Fréchet spaces ces(p+),1 <p< ∞

Albanese

Bonet

Ricker

2018

Journal of Mathematical Analysis and Applications

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The Banach spaces ces(p), 1 < p < ∞, were intensively studied by G. Bennett and others. The largest solid Banach lattice in C N which contains ℓ p and which the Cesàro operator C : C N −→ C N maps into ℓ p is ces(p). For each 1 ≤ p < ∞, the (positive) operator C also maps the Fréchet space ℓ p+ = ∩ q>p ℓ q into itself. It is shown that the largest solid Fréchet lattice in C N which contains ℓ p+ and which C maps into ℓ p+ is precisely ces(p+) := ∩ q>p ces(q). Although the spaces ℓ p+ are well understood, it seems that the spaces ces(p+) have not been considered at all. A detailed study of the Fréchet spaces ces(p+), 1 ≤ p < ∞, is undertaken. They are very dierent to the Fréchet spaces ℓ p+ which generate them in the above sense. We prove that each ces(p+) is a power series space of nite type and order one, and that all the spaces ces(p+), 1 ≤ p < ∞, are isomorphic.

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Section: Further Properties Of Ces(p+)mentioning

confidence: 99%

The Fréchet spaces ces(p+),1 <p< ∞

Albanese

Bonet

Ricker

2018

Journal of Mathematical Analysis and Applications

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“…such that inf i∈I0 a n (i)/a n * (i) = 0, [6,Theorem 18]. Since λ ∞ (A) is non-normable, the argument in the proof of [9,Corollary 2.4] shows that the increasing sequence (I m ) m∈N given by Condition D satisfies ∞ m=1 I m = I, but that no I m coincides with I. Suppose there exists x ∈ λ ∞ (A) such that {xχ Im } ∞ m=1 fails to converge to x.…”

Section: Proposition 41 Let X Be a Fréchet Space Which Is A Grothenmentioning

confidence: 99%

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

Bonet

Ricker

2006

Positivity

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It is shown that every echelon space λ∞(A), with A an arbitrary Köthe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ∞(A) is Montel if and only if it coincides with λ0(A), this identifies an extensive class of non-normable, non-Montel Fréchet spaces having these two properties. Even though the canonical unit vectors in λ∞(A) fail to form an unconditional basis whenever λ∞(A) = λ0(A), it is shown, nevertheless, that in this case λ∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given. Mathematics Subject Classification (2000). Primary 46A45, 46G10, 47B37; Secondary 46A04, 46A11, 46A35.

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“…If, in addition, P is σ -additive in L b (X ), then P is called boundedly σ -additive. For examples of spectral measures in classical spaces, some of which are boundedly σ -additive and others which are not, we refer to [9][10][11][12]34,35], for instance. For explicit examples of spaces which admit spectral measures which fail to be boundedly σ -additive we refer to Remark 4.3 in [1].…”

Section: Remark 312mentioning

confidence: 99%

Grothendieck spaces with the Dunford–Pettis property

2009

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Banach spaces which are Grothendieck spaces with the Dunford-Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11:77-93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For p ∈ {0} ∪ [1, ∞) it is shown that the classical co-echelon spaces k p (V ) and K p (V ) are GDP-spaces if and only if they are Montel. On the other hand, K ∞ (V ) is always a GDP-space and k ∞ (V ) is a GDP-space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ 1 (A), is distinguished.

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The Canonical Spectral Measure in Köthe Echelon Spaces

Abstract: Operator and measure theoretic properties of the canonical spectral measure acting in Köthe echelon sequence spaces X are characterized via topological and geometric properties of X (such as being nuclear, Montel, satisfying the density condition, etc.).

Cited by 9 publications

References 15 publications

The Fréchet spaces ces(p+),1 <p< ∞

The Fréchet spaces ces(p+),1 <p< ∞

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

Grothendieck spaces with the Dunford–Pettis property

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