The Fatou Completion of a Fréchet Function Space and Applications

Campo, Ricardo del; Ricker, Werner J.

doi:10.1017/s1446788709000238

Cited by 4 publications

(6 citation statements)

References 18 publications

(24 reference statements)

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“…We also mention`p + , 1 p < 1 (see [23]) and L p , 1 < p < 1 (see [11]). Further examples are L ) consisting of the p-th power m-integrable (respectively, weakly m-integrable) functions are Fréchet lattices (see [9], [10], [25]). …”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Further examples are L p loc (Ω) for 1 p ∞ with Ω ⊆ R N open and C(Ω), equipped with the topology of uniform convergence on compact subsets of Ω. Finally, if m is any Fréchet space-valued vector measure, then the spaces L p (m) (respectively, L p w (m)) consisting of the pth power m-integrable (respectively, weakly m-integrable) functions are Fréchet lattices [10,11,25].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Mean ergodic operators and reflexive Fréchet lattices

Bonet¹,

Pagter²,

Ricker³

2011

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood; see the works of Emel'yanov, Wol¤ and Zaharopol cited in the references. For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here (amongst other things) that every power bounded linear operator on E is mean ergodic if and only if E is re ‡exive if and only if E is Dedekind -complete and every positive power bounded operator on E is mean ergodic if and only if every positive power bounded operator in the strong dual E 0 (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria which detect when E admits a (positively) complemented lattice copy of c0,`1 or`1.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Mean ergodic operators and reflexive Fréchet lattices

Bonet¹,

Pagter²,

Ricker³

2011

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…It is clear that p X is an order ideal of L 0 and that p X is a Fréchet function space (briefly F.f.s) over pp´1, 1q, B, µq; see for example, [2, §2.3], [10] and the references therein. Observe that X is continuously embedded into p X via (4.2) together with the fact that the inclusion X Ď rT, Xs is continuous; see Lemma 2.6(ii).…”

Section: Then Every Functionmentioning

confidence: 99%

“…It becomes a F.f.s. for the topology of uniform convergence of indefinite integrals; for the precise definition and details see, for example, [2, §2.4], [10].…”

Section: Then Every Functionmentioning

confidence: 99%

Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces

Curbera

Okada

Ricker

2019

Quaestiones Mathematicae

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The finite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over p´1, 1q having nontrivial Boyd indices. For X " L p , 1 ă p ă 8, this operator has been intensively investigated since the 1940's (also under the guise of the "airfoil equation"). Recently, the extension and inversion of T : X Ñ X for more general X has been studied in [6], where it is shown that there exists a larger space rT, Xs, optimal in a well defined sense, which contains X continuously and such that T can be extended to a continuous linear operator T : rT, Xs Ñ X. The purpose of this paper is to continue this investigation of T via a consideration of the X-valued vector measure m X : A Þ Ñ T pχ A q induced by T and its associated integration operator f Þ Ñ ş 1 ´1 f dm X . In particular, we present integral representations of T : X Ñ X based on the L 1 -space of m X and other related spaces of integrable functions.

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“…Extensions to Fréchet function spaces have been studied e.g. in [4]. This paper studies topological duals of more general locally convex function spaces where the topology is generated by an arbitrary collection of seminorms satisfying the usual BFS axioms.…”

Section: Introductionmentioning

confidence: 99%

Topological duals of locally convex function spaces

Pennanen

Perkkiö

2022

Positivity

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This paper studies topological duals of locally convex function spaces that are natural generalizations of Fréchet and Banach function spaces. The dual is identified with the direct sum of another function space, a space of purely finitely additive measures and the annihilator of $$L^\infty $$ L ∞ . This allows for quick proofs of various classical as well as new duality results e.g. in Lebesgue, Musielak–Orlicz, Orlicz–Lorentz space and spaces associated with convex risk measures. Beyond Banach and Fréchet spaces, we obtain completeness and duality results in general paired spaces of random variables.

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The Fatou Completion of a Fréchet Function Space and Applications

Cited by 4 publications

References 18 publications

Mean ergodic operators and reflexive Fréchet lattices

Mean ergodic operators and reflexive Fréchet lattices

Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces

Topological duals of locally convex function spaces

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