An Application of Kadets-Pełczyński Sets to Narrow Operators

A known analogue of the Pitt compactness theorem for function spaces asserts that if 1 ≤ p < 2 and p < r < ∞, then every operator T : Lp → Lr is narrow. Using a technique developed by M.I. Kadets and A. Pełczyński, we prove a similar result. More precisely, if 1 ≤ p ≤ 2 and F is a Köthe {Ba...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2013
Автори:	Krasikova, I.V., Popov, M.M.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2013
Назва видання:	Журнал математической физики, анализа, геометрии
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/106739
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	An Application of Kadets-Pełczyński Sets to Narrow Operators / I.V. Krasikova, M.M. Popov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 102-107. — Бібліогр.: 14 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-106739
record_format	dspace
spelling	irk-123456789-1067392016-10-04T03:02:39Z An Application of Kadets-Pełczyński Sets to Narrow Operators Krasikova, I.V. Popov, M.M. A known analogue of the Pitt compactness theorem for function spaces asserts that if 1 ≤ p < 2 and p < r < ∞, then every operator T : Lp → Lr is narrow. Using a technique developed by M.I. Kadets and A. Pełczyński, we prove a similar result. More precisely, if 1 ≤ p ≤ 2 and F is a Köthe {Banach space on [0; 1] with an absolutely continuous norm containing no isomorph of Lp such that F is subset of Lp, then every regular operator T : Lp → F is narrow. Известный аналог теоремы Питта о компактности для функциональных пространств утверждает, что если 1 ≤ p < 2 и p < r < ∞, то каждый оператор Lp → Lr узкий. Используя технику, разработанную М.И. Кадецем и А. Пелчинским, мы доказываем похожий результат. Именно, если 1 ≤ p ≤ 2 и F - банахово пространство Кете на [0; 1] с абсолютно непрерывной нормой, не содержащее подпространств, изоморфных Lp, причем F является подмножеством Lp, то каждый регулярный оператор T : Lp → F узкий. 2013 Article An Application of Kadets-Pełczyński Sets to Narrow Operators / I.V. Krasikova, M.M. Popov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 102-107. — Бібліогр.: 14 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106739 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A known analogue of the Pitt compactness theorem for function spaces asserts that if 1 ≤ p < 2 and p < r < ∞, then every operator T : Lp → Lr is narrow. Using a technique developed by M.I. Kadets and A. Pełczyński, we prove a similar result. More precisely, if 1 ≤ p ≤ 2 and F is a Köthe {Banach space on [0; 1] with an absolutely continuous norm containing no isomorph of Lp such that F is subset of Lp, then every regular operator T : Lp → F is narrow.
format	Article
author	Krasikova, I.V. Popov, M.M.
spellingShingle	Krasikova, I.V. Popov, M.M. An Application of Kadets-Pełczyński Sets to Narrow Operators Журнал математической физики, анализа, геометрии
author_facet	Krasikova, I.V. Popov, M.M.
author_sort	Krasikova, I.V.
title	An Application of Kadets-Pełczyński Sets to Narrow Operators
title_short	An Application of Kadets-Pełczyński Sets to Narrow Operators
title_full	An Application of Kadets-Pełczyński Sets to Narrow Operators
title_fullStr	An Application of Kadets-Pełczyński Sets to Narrow Operators
title_full_unstemmed	An Application of Kadets-Pełczyński Sets to Narrow Operators
title_sort	application of kadets-pełczyński sets to narrow operators
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/106739
citation_txt	An Application of Kadets-Pełczyński Sets to Narrow Operators / I.V. Krasikova, M.M. Popov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 102-107. — Бібліогр.: 14 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT krasikovaiv anapplicationofkadetspełczynskisetstonarrowoperators AT popovmm anapplicationofkadetspełczynskisetstonarrowoperators AT krasikovaiv applicationofkadetspełczynskisetstonarrowoperators AT popovmm applicationofkadetspełczynskisetstonarrowoperators
first_indexed	2025-07-07T18:55:40Z
last_indexed	2025-07-07T18:55:40Z
_version_	1837015542488104960
fulltext	Journal of Mathematical Physics, Analysis, Geometry 2013, vol. 9, No. 1, pp. 102–107 An Application of Kadets–PeÃlczyński Sets to Narrow Operators I.V. Krasikova Department of Mathematics, Zaporizhzhya National University 66 Zhukows’koho Str., Zaporizhzhya, Ukraine E-mail: yudp@mail.ru M.M. Popov Department of Applied Mathematics, Chernivtsi National University 2 Kotsyubyns’koho Str., Chernivtsi 58012, Ukraine E-mail: misham.popov@gmail.com Received September 27, 2012 A known analogue of the Pitt compactness theorem for function spaces asserts that if 1 ≤ p < 2 and p < r < ∞, then every operator T : Lp → Lr is narrow. Using a technique developed by M.I. Kadets and A. PeÃlczyński, we prove a similar result. More precisely, if 1 ≤ p ≤ 2 and F is a Köthe–Banach space on [0, 1] with an absolutely continuous norm containing no isomorph of Lp such that F ⊂ Lp, then every regular operator T : Lp → F is narrow. Key words: narrow operator, Köthe function space, Banach space Lp. Mathematics Subject Classification 2010: 46A35 (primary), 46B15, 46A40, 46B42 (secondary). To the memory of M.I. Kadets 1. Introduction Narrow operators were introduced and studied by A.M. Plichko and the se- cond named author in [11]. Let us recall the definition for function spaces on the Lebesgue measure space ([0, 1], Σ, µ). Let L0 denote the linear space of all equivalence classes of Σ-measurable functions x : [0, 1] → R, and Lp = Lp[0, 1] for 1 ≤ p ≤ ∞. By 1A we denote the characteristic function of a set A ∈ Σ. We set Σ(A) = {B ∈ Σ : B ⊆ A}, Σ+(A) = {B ∈ Σ(A) : µ(B) > 0} and, as a partial case, Σ+ = Σ+([0, 1]). The notation A = B t C means that A = B ∪ C and B ∩ C = 0. By a sign we mean any {−1, 0, 1}-valued element x ∈ L0. More precisely, a sign x is called a sign on a set A ∈ Σ provided that suppx = A. c© I.V. Krasikova and M.M. Popov, 2013 An Application of Kadets–PeÃlczyński Sets to Narrow Operators A sign x is said to be of mean zero if ∫ [0,1] x dµ = 0. Observe that x ∈ L0 is a sign on A ∈ Σ if and only if x = 1B − 1C for some B,C ∈ Σ with A = B t C, and, in addition, µ(B) = µ(C) means that x is of mean zero. A Banach space E ⊂ L1 is called a Köthe–Banach space on [0, 1] if the following conditions hold: (1) 1[0,1] ∈ E; (2) for each x ∈ L0 and y ∈ E the condition \|x\| ≤ \|y\| implies x ∈ E and ‖x‖ ≤ ‖y‖. Note that, in the terminology of Lindenstrauss–Tzafriri [10, p. 28], a Köthe function space is a somewhat general notion which concerns the linear subspaces E of L0, because we additionally assume the inclusion E ⊆ L1. Using this inclu- sion and the closed graph theorem, one can show that the inclusion embedding of E to L1 is continuous. A further convenience of the integrability assumption E ⊆ L1 is shown in the following useful observation. Let E and F be Köthe– Banach spaces on [0, 1] with E ⊆ F . Then the inclusion embedding J : E → F , Jx = x for all x ∈ E, is continuous. Indeed, given any Köthe–Banach space G on [0, 1], by continuity of the inclusions G ⊆ L1 ⊆ L0 where the convergence in L0 is equivalent to the convergence in measure, we have that every convergent sequence in G converges in measure. Using this fact and the closed graph theorem, one can easily prove that any inclusion of Köthe–Banach spaces is continuous. A Köthe–Banach space E on [0, 1] is said to have an absolutely continuous norm if limµ(A)→0 ‖x · 1A‖ = 0 for every x ∈ E. By L(X, Y ) we denote the set of all linear bounded operators from a Banach space X to a Banach space Y , and set L(X) = L(X,X). Let E be a Köthe- Banach space on [0, 1] and let X be a Banach space. An operator T ∈ L(E, X) is called narrow if for every A ∈ Σ and every ε > 0 there is a mean zero sign x on A with ‖Tx‖ < ε. It is not very hard to show that if E has an absolutely continuous norm, then every compact operator T ∈ L(E,X) is narrow [11]. Thus, narrow operators generalize compact operators (as well as some other natural classes of “small” operators). Some properties of compact operators inherit by narrow operators, but not all of them (see [11], a recent survey [12] and a forthcoming book [13]). The classical Pitt theorem [9, p. 76] asserts that for any 1 ≤ p < r < ∞ every operator T ∈ L(`r, `p) is compact. Using the notion of infratype for Banach spaces, the following result was obtained in [8]. Theorem 1.1. If 1 ≤ p < 2 and p < r < ∞, then every operator T ∈ L(Lp, Lr) is narrow. Theorem 1.1 can be considered as an analogue of the Pitt compactness the- orem in the setting of function spaces. We remark that Theorem 1.1 is false for Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 103 I.V. Krasikova and M.M. Popov any other values of p and r. If p ≥ 2, then the composition Jr ◦ Ip of the identity embedding Ip : Lp → L2 and the isomorphic embedding Jr : L2 → Lr is evidently not narrow. And if 1 ≤ p < 2 and 1 ≤ r ≤ p, then the identity embedding of Lp into Lr is not narrow. Recall that a linear operator T : E → F between Köthe–Banach spaces (more general, between vector lattices) E and F is called positive if Tx ≥ 0 for every x ∈ E with x ≥ 0. Here and in sequel x ≤ y for elements of L1 means that x(t) ≤ y(t) holds a.e. on [0, 1]. A linear operator T : E → F is called regular if it is a difference of two positive linear operators from E to F . The main result of the paper is the following theorem. Theorem 1.2. Let 1 ≤ p ≤ 2 and let F be a Köthe–Banach space on [0, 1] with an absolutely continuous norm containing no subspace isomorphic to Lp such that F ⊂ Lp. Then every regular operator T ∈ L(Lp, F ) is narrow. Theorems 1.2 and 1.1 are incomparable: Theorem 1.2 covers much more range spaces, however it is restricted to regular operators. 2. Kadets–PeÃlczyński Sets In seminal paper [7] (1962), which became one of the most cited classical papers on the geometric theory of Banach spaces, M.I. Kadets and A. PeÃlczyński introduced special sets Mp ε in the space Lp, 1 ≤ p < ∞ depending on a positive parameter ε > 0 and consisting of all elements x ∈ Lp such that the subgraph of the decreasing rearrangement of \|x\| contains a square with sides ε. Let us give a precise definition for the general setting of the Köthe–Banach spaces on [0, 1]. Definition 2.1. Let E be a Köthe–Banach space on [0, 1] and ε > 0. Set ME ε = { x ∈ E : µ { t ∈ [0, 1] : \|x(t)\| ≥ ε‖x‖E } ≥ ε } . Obviously, ME ε′ ⊆ ME ε′′ whenever ε′ ≥ ε′′ and ⋃ ε>0 ME ε = E. Remark that the sets ME ε for the setting of the Köthe–Banach spaces were used by various authors, see, e.g., [10, Proposition 1, p. 8], [4, 5]. The idea of using these sets can be explained as follows. Given a normalized sequence (xn) in E, either it is contained in some universal set ME ε , or for every ε > 0 there is n such that xn /∈ ME ε . In the first case, the norm of E and the L1-norm are equivalent on (xn), and in the second case, (xn) contains subsequences with arbitrarily “narrow” elements. This leads to different interesting alternatives for the sequences and subspaces of E. One of the alternatives which we will need later is obtained in the following lemma (see [13, Lemma 10.63]; we provide its proof below for the sake of completeness). 104 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 An Application of Kadets–PeÃlczyński Sets to Narrow Operators Lemma 2.2. Let E be a Köthe–Banach space on [0, 1] with an absolutely continuous norm. Let (xn) be an order bounded sequence from E such that for every ε > 0 there exist n ∈ N such that xn /∈ ME ε . Then there exists a subsequence (yn) of (xn) and a disjoint sequence (zn) in E such that \|zn\| ≤ \|yn\| for all n, and ‖yn − zn‖ → 0. Before the proof, we recall some lattice definitions. A subset X of a Köthe– Banach space E is called order bounded provided there exists y ∈ E such that \|x\| ≤ y for each x ∈ X. A linear operator T : E → F between Köthe–Banach spaces E and F is called order bounded if T sends order bounded sets from E to order bounded sets in F . Evidently, any positive operator (hence, any regular operator) is order bounded. By E+ we denote the positive cone of E, that is, E+ = {x ∈ E : x ≥ 0}. P r o o f. Let e ∈ E+ be such that \|xn\| ≤ e for all n ∈ N. Choose a subsequence (x′n) of (xn) so that x′n /∈ ME 2−n for all n. For every n ∈ N, let An = {t ∈ [0, 1] : \|x′n(t)\| ≥ 2−n‖x′n‖} and Bn = ⋃∞ k=n Ak. Note that µ(An) < 2−n, Bn+1 ⊆ Bn, and µ(Bn) ≤ 2−n+1 for each n. Choose a strictly increasing sequence of the integers (ni)i such that ‖e · 1Bni+1 ‖ ≤ 1/i (this is possible because of the absolute continuity of the norm). Observe that the sets Ci = Ani \ Bni+1 are disjoint. Let yi = x′ni and zi = yi · 1Ci for i = 1, 2, . . .. Then (zi) is a disjoint sequence, \|zi\| ≤ \|yi\|, and ‖yi − zi‖ = ‖x′ni · 1[0,1]\Ci ‖ ≤ ‖x′ni · 1[0,1]\Ani ‖+ ‖x′ni · 1Bni+1 ‖ ≤ ‖2−ni‖x′ni ‖ · 1[0,1]\Ani ‖+ ‖e · 1Bni+1 ‖ ≤ 2−ni‖e‖‖1[0,1]‖+ 1/i → 0 as i →∞. We need the following lemma which in a certain degree develops the previous one. Lemma 2.3. Let E be a Köthe–Banach space on [0, 1] with an absolutely continuous norm. Let (xn) be an order bounded sequence from E such that ‖xn‖ ≥ δ for some δ > 0 and all n ∈ N. Then there exists ε > 0 such that xn ∈ ME ε for all n. P r o o f. Let y ∈ E be such that \|xn\| ≤ y for all n ∈ N. Supposing the lemma is false, choose by Lemma 2.2 a subsequence (yn) of (xn) and a disjoint sequence (zn) in E such that \|zn\| ≤ \|yn\| for all n, and ‖yn − zn‖ → 0. Set An = supp zn for each n ∈ N. Then \|zn\| ≤ \|yn\| · 1An ≤ z · 1An and hence δ ≤ ‖yn‖ ≤ ‖zn‖+ ‖yn − zn‖ ≤ ∥∥z · 1An ∥∥ + ‖yn − zn‖ for all n. This is impossible, because ‖yn − zn‖ → 0 and ∥∥z · 1An ∥∥ → 0 by the absolute continuity of the norm in E. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 105 I.V. Krasikova and M.M. Popov 3. Enflo Operators and Proof of the Main Result Let X be a Banach space. An operator T ∈ L(X) is called an Enflo operator if there is a subspace Y of X isomorphic to X such that the restriction T \|Y of T to Y is an isomorphic embedding. The name “Enflo operator” is due to the following famous Enflo theorem on primarity of Lp: if the space Lp, 1 ≤ p < ∞, is decomposed into a direct sum of closed subspaces Lp = X ⊕ Y , then at least one of X,Y is isomorphic to Lp (see [10, p. 179]). One of the peculiarities of the spaces Lp with 1 ≤ p < 2, which will be used later, is described in the following deep theorem due to varios authors. Theorem 3.1. Let 1 ≤ p ≤ 2. Then any non-Enflo operator T ∈ L(Lp) is narrow. Theorem 3.1 for p = 1 can be deduced from the results of [3]. Moreover, the following remarkable result of Rosenthal (the equivalence of (c) and (d) in Theorem 1.5 of [14]) gives much more — necessary and sufficient conditions for an operator T ∈ L(L1) to be narrow. Theorem 3.2. An operator T ∈ L(L1) is narrow if and only if for each A ∈ Σ the restriction T ∣∣ L1(A) is not an isomorphic embedding, where L1(A) = {x ∈ L1 : suppx ⊆ A}. Theorem 3.1 in the case 1 < p < 2 was announced by J. Bourgain [1, The- orem 4.12, item 2, p. 54] without a proof, accompanied with a citation to [6]. Formally, Theorem 3.1 cannot be deduced from [6], however an involved proof can be written by using the ideas and methods of [6] (such a proof is to be found in [13, Section 7.3]). Another proof of Theorem 3.1 in the case 1 < p < 2 is given in [2]. The last our comment is that for p = 2 Theorem 3.1 holds trivially, because in this case a non-Enflo operator must be compact and hence narrow. Now we are ready to prove the main result. P r o o f of Theorem 1.2. Denote by J ∈ L(F, Lp) the inclusion embedding, Jx = x for all x ∈ F . Consider any T ∈ L(Lp, F ) and assume, on the contrary, that T is not narrow. Choose A ∈ Σ+ and δ > 0 such that ‖Tx‖ ≥ δ for each mean zero sign x on A. Set S = J ◦ T ∈ L(Lp) and show that S is non-narrow as well. Assuming, on the contrary, that S is narrow, we find a sequence (xn) of mean zero signs on A such that ‖Sxn‖ → 0. Since (xn) is order bounded by 1[0,1] and T is regular, (Txn) is an order bounded sequence. And since ‖Txn‖ ≥ δ for all n ∈ N, by Lemma 2.3, there exists ε > 0 such that Txn ∈ MF ε for all n. Thus, ‖Sxn‖p p = ∫ [0,1] \|Txn\|pdµ ≥ εp+1, 106 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 An Application of Kadets–PeÃlczyński Sets to Narrow Operators which contradicts the condition ‖Sxn‖ → 0. Thus, S is non-narrow. By The- orem 1.2, S is an Enflo operator. Let E be a subspace of Lp isomorphic to Lp such that ‖Sx‖ ≥ α‖x‖ for some α > 0 and all x ∈ E. Then ‖Tx‖ ≥ ‖J‖−1‖Sx‖ ≥ α‖J‖−1‖x‖ for all x ∈ E which contradicts the assumption that F contains no subspace isomorphic to Lp. We do not know whether the assumption of the regularity of T is essential in Theorem 1.2. References [1] J. Bourgain, New Classes of Lp-spaces. — Lect. Notes Math. 889 (1981), 1–143. [2] D. Dosev, W.B. Johnson, and G. Schechtman, Commutators on Lp, 1 ≤ p < ∞. — J. Amer. Math. Soc. 26 (2013), No. 1, 101–127. [3] P. Enflo and T. Starbird, Subspaces of L1 Containing L1. — Studia Math. 65 (1979), No. 2, 203–225. [4] J. Flores, F.L. Hernández, and P. Tradacete, Domination Problems for Strictly Singular Operators and Other Related Classes. — Positivity 15 (2011), No. 4, 595–616. [5] J. Flores and C. Ruiz, Domination by Positive Narrow Operators. — Positivity 7 (2003), No. 4, 303–321. [6] W.B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric Structures in Banach Spaces. — Mem. Amer. Math. Soc. 19 (217) (1979). [7] M.I. Kadets and A. PeÃlczyński, Bases, Lacunary Sequences and Complemented Subspaces in the Spaces Lp. — Studia Math. 21 (1962), No. 2, 161–176. [8] M.I. Kadets and M.M. Popov, On the Lyapunov Convexity Theorem with Applica- tions to Sign-Embeddings. — Ukr. Mat. Zh. 44 (1992), No. 9, 1192–1200. [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. 1, Sequence Spaces. Springer–Verlag, Berlin–Heidelberg–New York, 1977. [10] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. 2, Sequence Spaces. Springer–Verlag, Berlin–Heidelberg–New York, 1979. [11] A.M. Plichko and M.M. Popov, Symmetric Function Spaces on Atomless Probability Spaces. — Dissertationes Math. (Rozprawy Mat.) 306 (1990), 1–85. [12] M.M. Popov, Narrow Operators (a survey). — Banach Center Publ. 92 (2011), 299–326. [13] M.M. Popov and B. Randrianantoanina, Narrow Operators on Function Spaces and Vector Lattices. De Gruyter Studies in Mathematics 45, De Gruyter, 2012. [14] H.P. Rosenthal, Embeddings of L1 in L1. — Contemp. Math. 26 (1984), 335–349. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 107

An Application of Kadets-Pełczyński Sets to Narrow Operators

Репозитарії

Схожі ресурси