A property of non-strongly regular operators

Argyros, Spiros A.; Petrakis, Minos

doi:10.1017/cbo9780511662317.003

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“…Proof. (a) ⇒ (b) If H is representable, then we can find an essentially bounded measurable map ψ : Ω × [0, 1] → X that represents H. The map ψ ′ : [0, 1] → L 1 (λ, X) given by t → ψ(., t) belongs to L ∞ ([0, 1], L 1 (λ, X)): in fact ||ψ ′ (t)|| = Ω ||ψ(ω, t)|| dλ(ω) for every t ∈ [0, 1] hence ||ψ ′ || ∞ ≤ ||ψ|| ∞ and we claim that ψ ′ represents K: for each g ∈ L 1 [0, 1], 1], X) be the representing measure for S(ω) (i.e S(ω)(χ A ) = µ ω (A)). It is well known that S(ω) is representable if and only if µ ω has Bochner density with respect to dt.…”

Section: Theorem 1 [16](kalton) Suppose Thatmentioning

confidence: 97%

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The near Radon-Nikodym property in Lebesgue-Bochner function spaces

Randrianantoanina

Saab

1998

Illinois J. Math.

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Let X be a Banach space and (Ω, Σ, λ) be a finite measure space, 1 ≤ p < ∞. It is shown that L p (λ, X) has the Near Radon-Nikodym property if and only if X has it. Similarly if E is a Köthe function space that does not contain a copy of c 0 , then E(X) has the Near Radon-Nikodym property if and only if X does. 1991 Mathematics Subject Classification. 46E40, 46G10; Secondary 28B05, 28B20. Key words and phrases. Lebesgue-Bochner spaces, Representable operators. DEFINITIONS AND PRELIMINARY RESULTSThroughout this note, I n,k = [ k−1 2 n , k 2 n ) will be the sequence of dyadic intervals in [0, 1] and Σ n is the σ-algebra generated by the finite sequence (I n,k ) k=1,2 n . The word operator will always mean linear bounded operator and L(E, F ) will stand for the space of all operators from E into F . For any given Banach space E, its closed unit ball will be denoted by E 1 . Definition 1. Let X be a Banach space. An operator T : L 1 [0, 1] → X is said to be representable if there is a Bochner integrable function g ∈ L ∞ ([0, 1], X) such that T(f )= f g dµ for all f in L 1 [0, 1]. Definition 2. An operator D : L 1 [0, 1] → X is called a Dunford-Pettis operator if D sends weakly compact sets into norm compact sets.It is well known ([7] Example 5,III.2.11) that all representable operators from L 1 [0, 1] are Dunford-Pettis; but the converse is not true in general.Definition 3. An operator T : L 1 [0, 1] → X is said to be nearly representable if for each Dunford-Pettis operator D :The notion of nearly representable operators was introduced by Kaufman, Petrakis, Riddle and Uhl in [17]. It should be noted that since the class of Dunford-Pettis operators from L 1 [0, 1] into L 1 [0, 1] is a Banach lattice ([3]), if an operator T ∈ L(L 1 [0, 1], X) fails to be nearly representable then one can find a positive Dunford-Pettis operator D ∈ L(L 1 [0, 1], L 1 [0, 1]) such that T • D is not representable.[26] P. Wojtaszczyk. Banach spaces for analysts . Cambridge University Press, first edition, (1991).

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Section: Theorem 1 [16](kalton) Suppose Thatmentioning

confidence: 97%

“…Examples of Banach spaces with the NRNP are spaces with the RNP, L 1 -spaces, L 1 /H 1 . For more detailed discussion on the NRNP and nearly representable operators, we refer to [1], [11] and [17].…”

Section: Introductionmentioning

confidence: 99%

The near Radon-Nikodym property in Lebesgue-Bochner function spaces

Randrianantoanina

Saab

1998

Illinois J. Math.

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“…Epeid ènac q¸roc X èqei thn RNP an kai mìno an kje telest c ston L(L 1 , X) eÐnai anaparastsimoc, to parapnw je¸rhma deÐqnei ìti an kje Dunford-Pettis telest c eÐnai anaparastsimoc, tìte o X èqei thn RNP. To 1990 oi S. Argurìc kai M. Petrkhc sto rjro touc A property of non-strongly regular operators ( [7]) apodeiknÔoun apotèlesma antÐstoiqo tou jewr matoc 3.14, gia non-strongly regular telestèc. Eidikìtera: Jewrhma 3.16.…”

Section: Mia Anaparstash Tou Sunìlou 1ωmentioning

confidence: 99%

“…Mèsw autoÔ tou jewr matoc kai tou apotelèsmatoc tou Schachermayer prokÔptei: Porisma 3.17. (Argurìc -Petrkhc, 1990, Cor.1, [7]) 'Estw K kleistì, kurtì uposÔnolo enìc q¸rou Banach X. Upojètoume ìti gia ton telest T : L 1 (0, 1) → X me T (P) ⊂ K (ìpou P to sÔnolo twn densities ston L 1…”

Section: Mia Anaparstash Tou Sunìlou 1ωmentioning

confidence: 99%

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Μελέτη γεωμετρικών ιδιοτήτων χώρων Banach

Pavlakos¹,

Παυλάκος²

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Αφιερώνεται στον Μίνωα, την Αρετή και την Κλέα v Prìlogoc H paroÔsa Didaktorik Diatrib apoteleÐ thn katagraf twn apotelesmtwn thc ereunhtik c mou melèthc, gewmetrik¸n idiot twn q¸rwn Banach, upì thn kajod ghsh kai epÐbleyh tou Dasklou mou, EpÐkourou Kajhght sto PoluteqneÐo Kr thc, k. MÐnwa Petrkh. H pro-Perieqìmena EISAGWGH xiii Keflaio 1. STOIQEIA APO THN JEWRIA QWRWN BANACH 1.1. BasikoÐ orismoÐ, eujÔ ginìmeno, probolèc, • -topologÐa kai w-topologÐa, Je¸rhma Mazur 1.2. Bseic kai basikèc akoloujÐec, probolèc merik¸n ajroismtwn, isodunamÐa bsewn, block basikèc akoloujÐec, Principle of small pertrubations, arq thc epilog c twn Bessaga-Pelzynski, bseic se klasikoÔc q¸rouc, sÔsthma Haar 1.3. Unconditional, shrinking kai boundedly complete bseic, ènac qarakthrismìc gia touc autopajeÐc q¸rouc 1.4. DeÐktec, dèndra, d-bushes, d-approximate bushes, averaging back bushes 1.5. To arqètupo (fundamental) bush ston c 0 Keflaio 2. RADON-NIKODYM, KREIN-MILMAN KAI SUNAFEIS IDIOTHTES 2.1. Dianusmatik mètra, metr simec sunart seic, olokl rwma Bochner kai anaparastsimoi telestèc (RNP mèroc 1o) 2.2. Dianusmatik martingales kai dentability (RNP mèroc 2o) 2.3. Idiìthta Krein-Milman, h RNP sunepgetai thn KMR 2.4. H RNP eÐnai isodÔnamh me thn KMR stouc diaqwrÐsimouc duikoÔc (1975), sta Banach lattices (1981), ìtan o q¸roc eÐnai isìmorfoc me to tetrgwnì tou (1985) 2.5. H RNP eÐnai isodÔnamh me thn KMR sta strongly regular sÔnola (1985), sta uposÔnola q¸rwn me unconditional bsh (1985), point of continuity property 2.6. H RNP eÐnai isodÔnamh me thn KMR ìtan h asjen c kai h norm topologÐa sumpÐptoun (1989), sta non-dentable non-PCP uposÔnola q¸rwn me unconditional skipped block decomposition, denting points, an o q¸roc den èqei thn RNP tìte perièqei upìqwro me FDD pou den èqei thn RNP ix x PERIEQ OMENA 2.7. CFDSD, Pal representations, mia efarmog ston c 0 , h RNP eÐnai isodÔnamh me thn KMR sta uposÔnola tou jetikoÔ k¸nou tou L 1 (0, 1) (1993) Keflaio 3. KAJE NON-DENTABLE UPOSUNOLO TOU C(ω ω k ) PERIEQEI ENA KLEISTO KURTO UPOSUNOLO L TETOIO WSTE TO L EQEI THN PCP KAI OQI THN RNP. SUNEPWS H RNP KAI H KMR EINAI ISODUNAMES IDIOTHTES STA UPOSUNOLA TWN QWRWN BANACH C(ω ω k ) 3.1. Oi q¸roi Banach C(α) ìpou α arijm simoc diataktikìc arijmìc, h kathgoriopoÐhsh twn C(K) me K sumpag metrikì q¸ro (1960 kai 1966), mia sunèpeia ìtan o q¸roc Banach X den perièqei ton 1 (1978) 3.2. O Cantor-Bendixson deÐkthc, parathr seic, gnwst apotelèsmata gia K arijm simo sumpagèc 3.3. Mia anaparstash tou sunìlou 1,ω ω , o q¸roc C(ω ω ) den emfuteÔetai se q¸ro me unconditional bsh 3.4. Megloi telestèc ston L 1 (0, 1) me mikrèc probolèc, telestèc Dunford-Pettis 3.5. Block δ-approximate bushes, sunèpeiec thc unconditionality (ajroismtwn) se q¸rouc me Schauder dimensional decomposition 3.6. Sunèpeiec ìtan K kleistì, kurtì, fragmèno, non-PCP, uposÔnolo tou X kai o q¸roc Banach X den perièqei ton xvi EISAGWGH small combination of slices idiìthta. Tìte uprqei èna kleistì, kurtì, mh kenì uposÔnolo W tou K tètoio¸ste : ( * ) to W eÐnai non-dentable kai h norm me thn asjen topolo...

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A property of non-strongly regular operators

Cited by 2 publications

References 0 publications

The near Radon-Nikodym property in Lebesgue-Bochner function spaces

The near Radon-Nikodym property in Lebesgue-Bochner function spaces

Μελέτη γεωμετρικών ιδιοτήτων χώρων Banach

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