Wyniki wyszukiwania

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Lattice copies of c₀ and $ℓ^{∞}$ in spaces of integrable functions for a vector measure

100%

Okada S., Ricker W., Pérez E.

Instytut Matematyczny Polskiej Akademii Nauk

EN

The spaces L¹(m) of all m-integrable (resp. $L¹_{w}(m)$ of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, $L¹_{w}(m)$ is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If, additionally, the inclusion $L¹(m) ⊆ L¹_{w}(m)$ (which always holds) is proper, then L¹(m) and $L¹_{w}(m)$ contain lattice-isomorphic copies of the complex Banach lattices c₀ and $ℓ^∞$, respectively. On the other hand, whenever L¹(m) contains an isomorphic copy of c₀, merely in the lcHs sense, then necessarily $L¹(m) ⊊ L¹_{w}(m)$. Moreover, the X-valued integration operator Iₘ: f ↦ ∫ fdm, for f ∈ L¹(m), then fixes a copy of c₀. For X a Banach space, the validity of $L¹(m) = L¹_{w}(m)$ turns out to be equivalent to Iₘ being weakly completely continuous. A sufficient condition for this is the (q,1)-concavity of Iₘ for some 1 ≤ q < ∞. This criterion is fulfilled when Iₘ belongs to various classical operator ideals. Unlike for $L¹_{w}(m)$, the space L¹(m) can never contain an isomorphic copy of $ℓ^{∞}$. A rich supply of examples and counterexamples is presented. The methods involved are a hybrid of vector measure/integration theory, functional analysis, operator theory and complex vector lattices.

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Sequential closedness of Boolean algebras of projections in Banach spaces

92%

Fremlin D., de Pagter B., Ricker W.

Studia Mathematica

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2005

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tom 167

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nr 1

45-62

EN

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a σ-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).

3

Compactness of the integration operator associated with a vector measure

92%

Okada S., Ricker W., Rodríguez-Piazza L.

Studia Mathematica

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2002

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tom 150

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nr 2

133-149

EN

A characterization is given of those Banach-space-valued vector measures m with finite variation whose associated integration operator Iₘ: f ↦ ∫fdm is compact as a linear map from L¹(m) into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures m (with finite variation) such that Iₘ is compact, and other m (still with finite variation) such that Iₘ is not compact. If m has infinite variation, then Iₘ is never compact.

Ograniczanie wyników

2 Studia Mathematica

3 Ricker W.

2 Okada S.

1 Fremlin D.

1 Pérez E.

1 Rodríguez-Piazza L.

1 de Pagter B.

1 2014

1 2005

1 2002

Lattice copies of c₀ and $ℓ^{∞}$ in spaces of integrable functions for a vector measure

Sequential closedness of Boolean algebras of projections in Banach spaces

Compactness of the integration operator associated with a vector measure