DOCSLIB.ORG

Grothendieck Spaces, Operators, and Beyond

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Grothendieck Spaces, Operators, and Beyond Grothendieck spaces, operators, and beyond Tomasz Kania Academy of Sciences of the Czech Republic, Praha Zimní škola z abstraktní analýzy Svratka, 13 leden 2019 joint work with K. Beanland & N. J. Laustsen 1 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I call such spaces Grothendieck. This class is I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. 2 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I call such spaces Grothendieck. This class is I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. 2 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is 2 I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! 2 I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), 2 I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! 2 s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. 2 So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. 2 Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... 2 C(K) for K Stonean, σ-Stonean, F -space, all indecomposable spaces C(K) constructed by Koszmider, Plebanek, and others. No topological characterisation of when a C(K)-space is Grothendieck exists. If C(K) is Grothendieck, K does not have non-trivial convergent sequences. 1 H (D), bdd analytic functions (Bourgain 1983) Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space. Grothendieck spaces are rare animals Some examples. 3 1 H (D), bdd analytic functions (Bourgain 1983) Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space. Grothendieck spaces are rare animals Some examples. C(K) for K Stonean, σ-Stonean, F -space, all indecomposable spaces C(K) constructed by Koszmider, Plebanek, and others. No topological characterisation of when a C(K)-space is Grothendieck exists. If C(K) is Grothendieck, K does not have non-trivial convergent sequences. 3 Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space.
Load more

Recommended publications
  • Topology Proceedings
    Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 26, 2001{2002 Pages 695{707 WEAKLY EBERLEIN COMPACT SPACES DANIEL JARDON´ ∗ Abstract. Call a space X weakly splittable if, for each f X 2 R , there exists a σ-compact F Cp(X) such that f F (the bar denotes the closure in RX⊂). A weakly splittable com-2 pact space is called weakly Eberlein compact. We prove that weakly Eberlein compact spaces have almost the same prop- erties as Eberlein compact spaces. We show that any weakly Eberlein compact space of cardinality 6 c is Eberlein com- pact. We prove that a compact space X is weakly Eberlein compact if and only if X is splittable over the class of Eber- lein compact spaces and that every countably compact weakly splittable space has the Preiss{Simon property. 0. Introduction The first one to study weakly compact subspaces of Banach spaces was Eberlein [Eb]. His results showed that these compact spaces are very important and have numerous applications in many areas of mathematics. That is why they were called Eberlein com- pact spaces. In fact, a compact space X is Eberlein compact if and only if Cp(X) has a σ-compact dense subspace and it is a non-trivial theorem that these two definitions are equivalent. The basic results of the theory of Eberlein compact spaces have many 2000 Mathematics Subject Classification.
    [Show full text]
  • Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed
    Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed To cite this version: Nasiruddin Ahmed. Weak Compactness in the Space of Operator Valued Measures and Optimal Control. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.49-58, 10.1007/978-3-642-36062-6_5. hal-01347522 HAL Id: hal-01347522 https://hal.inria.fr/hal-01347522 Submitted on 21 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License WEAK COMPACTNESS IN THE SPACE OF OPERATOR VALUED MEASURES AND OPTIMAL CONTROL N.U.Ahmed EECS, University of Ottawa, Ottawa, Canada Abstract. In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces. Keywords: Space of Operator valued measures, Countably additive op- erator valued measures, Weak compactness, Semigroups of bounded lin- ear operators, Optimal Structural control.
    [Show full text]
  • Approximation Boundedness Surjectivity
    APPROXIMATION BOUNDEDNESS SURJECTIVITY Olav Kristian Nygaard Dr. scient. thesis, University of Bergen, 2001 Approximation, Boundedness, Surjectivity Olav Kr. Nygaard, 2001 ISBN 82-92-160-08-6 Contents 1Theframework 9 1.1 Separability, bases and the approximation property ....... 13 1.2Thecompletenessassumption................... 16 1.3Theoryofclosed,convexsets................... 19 2 Factorization of weakly compact operators and the approximation property 25 2.1Introduction............................. 25 2.2 Criteria of the approximation property in terms of the Davis- Figiel-Johnson-Pe'lczy´nskifactorization.............. 27 2.3Uniformisometricfactorization.................. 33 2.4 The approximation property and ideals of finite rank operators 36 2.5 The compact approximation property and ideals of compact operators.............................. 39 2.6 From approximation properties to metric approximation prop- erties................................. 41 3 Boundedness and surjectivity 49 3.1Introduction............................. 49 3.2Somemorepreliminaries...................... 52 3.3 The boundedness property in normed spaces . ....... 57 3.4ThesurjectivitypropertyinBanachspaces........... 59 3.5 The Seever property and the Nikod´ymproperty......... 63 3.6 Some results on thickness in L(X, Y )∗ .............. 63 3.7Somequestionsandremarks.................... 65 4 Slices in the unit ball of a uniform algebra 69 4.1Introduction............................. 69 4.2Thesliceshavediameter2..................... 70 4.3Someremarks...........................
    [Show full text]
  • Grothendieck Spaces with the Dunford–Pettis Property
    GROTHENDIECK SPACES WITH THE DUNFORD–PETTIS PROPERTY ANGELA A. ALBANESE*, JOSÉ BONET+ AND WERNER J. RICKER Abstract. Banach spaces which are Grothendieck spaces with the Dunford– Pettis property (briefly, GDP) are classical. A systematic treatment of GDP– Fréchet spaces occurs in [12]. 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 kp(V ) and Kp(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 λ1(A), is distinguished. 1. Introduction. Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) play a prominent role in the theory of Banach spaces and vector measures; see Ch.VI of [17], especially the Notes and Remarks, and [18]. Known examples include L∞, ∞ ∞ H (D), injective Banach spaces (e.g. ` ) and certain C(K) spaces. D. Dean showed in [14] that a GDP–space does not admit any Schauder decomposition; see also [26, Corollary 8]. This has serious consequences for spectral measures in such spaces, [31]. For non–normable spaces the situation changes dramatically. Every Fréchet Montel space X is a GDP–space, [12, Remark 2.2]. Other than Montel spaces, the only known non–normable Fréchet space which is a GDP–space is the Köthe echelon space λ∞(A), for an arbitrary Köthe matrix A, [12, Proposition 3.1].
    [Show full text]
  • Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
    [ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments.
    [Show full text]
  • Spaces of Compact Operators N
    Math. Ann. 208, 267--278 (1974) Q by Springer-Verlag 1974 Spaces of Compact Operators N. J. Kalton 1. Introduction In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather dif- ferent problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F.
    [Show full text]
  • Josefson–Nissenzweig Property for Cp-Spaces
    RACSAM https://doi.org/10.1007/s13398-019-00667-8 ORIGINAL PAPER Josefson–Nissenzweig property for Cp-spaces T. Banakh1,2 · J. K¸akol3,4 · W. Sliwa´ 5 Received: 6 March 2019 / Accepted: 2 April 2019 © The Author(s) 2019 Abstract The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K ) admits a quotient isomorphic to Banach spaces c or 2.Theaimof the paper is to study a natural variant of this result for the space C p(X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson– Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C p(X)-spaces. We prove: for a Tychonoff space X the space C p(X) satisfies the JNP if and only if C p(X) has a N N quotient isomorphic to c0 := {(xn)n∈N ∈ R : xn → 0} (with the product topology of R )if and only if C p(X) contains a complemented subspace isomorphic to c0. The last statement provides a C p-version of the Cembranos theorem stating that the Banach space C(K ) is not a Grothendieck space if and only if C(K ) contains a complemented copy of the Banach space c0 with the sup-norm topology. For a pseudocompact space X the space C p(X) has the JNP if and only if C p(X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C p(K ) containing a complemented subspace isomorphic to c0 is given.
    [Show full text]
  • Grothendieck C(K)-Spaces of Small Density
    Grothendieck C(K)-spaces of small density Damian Sobota Technische Universit¨at Wien, Austria Winter School, Hejnice 2018 X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X X ∗∗ — the bidual space of X = the dual space of X ∗ Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X . Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; weak* topology on X ∗ = the smallest topology such that every evx is continuous; Weak topologies X — an infinite-dimensional Banach space X ∗∗ — the bidual space of X = the dual space of X ∗ Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X . Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; weak* topology on X ∗ = the smallest topology such that every evx is continuous; Weak topologies X — an infinite-dimensional Banach space X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X . Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; weak* topology on X ∗ = the smallest topology such that every evx is continuous; Weak topologies X — an infinite-dimensional Banach space X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X X ∗∗ — the bidual space of X = the dual space of X ∗ ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X .
    [Show full text]
  • Arxiv:2102.03838V2 [Math.FA] 4 Mar 2021
    GROTHENDIECK SPACES: THE LANDSCAPE AND PERSPECTIVES MANUEL GONZÁLEZ AND TOMASZ KANIA Abstract. In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak- and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr. Vector measures. Addi- tionally, Diestel’s paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day. The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describ- ing the main examples of spaces with this property, recording the solutions to Diestel’s problems, providing generalisations/extensions or new proofs of various results concern- ing Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory. arXiv:2102.03838v3 [math.FA] 7 Jul 2021 Date: July 8, 2021. 2010 Mathematics Subject Classification. Primary: 46A35; 46B20. Key words and phrases. Grothendieck space, G-space, property (V ), property (V1), Banach space, Gro- thendieck operator, C*-algebra, Banach lattice, positive Grothendieck property, quantitative Grothendieck property, pseudo-intersection number, locally convex Grothendieck space, twisted sum, push-out. Research of M. González was partially supported by MICINN (Spain), Grant PID2019-103961GB-C22. T. Kania acknowledges with thanks funding received from SONATA 15 No. 2019/35/D/ST1/01734. 1 2 M.
    [Show full text]
  • The Grothendieck Property of Weak L Spaces and Marcinkiewicz Spaces
    TU DELFT MASTER THESIS APPLIED MATHEMATICS The Grothendieck property of Weak Lp spaces and Marcinkiewicz spaces Supervisor Author Prof. Dr. B. DE PAGTER P. VAN DEN BOSCH 28-1-2019 Introduction In this master thesis the proof of Lotz in [40] that Weak Lp spaces have the Grothendieck property is studied. The proof is slightly modified to be more explicit and easier to comprehend by introducing lemma’s to better separate different parts of the proof that more clearly reveal its structure. Fur- thermore, the more general Marcinkiewicz spaces are shown to sometimes have the Grothendieck property, using the sufficient conditions for a Banach lattice to have the Grothendieck property that Lotz derived in [40] to prove the Grothendieck property of Weak Lp spaces. For most of these conditions that together are sufficient, proving that Marcinkiewicz spaces satisfy them is done in a way very similar to the case of Weak Lp spaces. However, the proof of the (necessary) condition that the dual sometimes has order continuous norm does not allow for such a simple generalization and requires more work. Finally, by using some more recent results [19] about the existence of symmetric functionals in the dual, the conditions that are given for the Grothendieck property of Marcinkiewicz spaces are shown to be necessary, and we thereby obtain a characterization of the Marcinkiewicz spaces that have the Grothendieck property. Chapter 1 briefly summarizes well-known theory about Banach spaces required in the subsequent chapters: the weak topology, the Grothendieck property and characterizations thereof. In chapter 2 Banach lattices are introduced and Lotz’ sufficient conditions for the Grothendieck property of a Banach lattice are derived.
    [Show full text]
  • 17 Jan 2021 Grothendieck-Type Subsets of Banach Lattices
    Grothendieck-type subsets of Banach lattices Pablo Galindo∗ and V. C. C. Miranda † Abstract In the setting of Banach lattices the weak (resp. positive) Grothendieck spaces have been defined. We localize such notions by defining new classes of sets that we study and compare with some quite related different classes. This allows us to introduce and compare the corresponding linear operators. Keywords: almost Grothendieck sets, almost Grothendieck operators, disjoint operators, positive Grothendieck property, positive Grothendieck sets, weak Grothendieck property Mathematics Subject Classification (2010) 46B42, 47B65. 1 Introduction and preliminaries Throughout this paper X and Y will denote Banach spaces, E and F will denote Banach lattices. We denote by BX the closed unit ball of X. Recall that X is a Grothendieck space if weak* null sequences in X′ are weakly null. In a Banach lattice, two further Grothendieck properties have been considered. A Banach lattice E has the weak Grothendieck property (resp. positive Grothendieck property) if every disjoint weak* null sequence in E′ is weakly null (resp. every positive weak* null sequence in E′ is weakly null ). We refer to [21] and [15] for definitions and some results concerning those two properties. A subset A of a Banach space X is said to be Grothendieck if T (A) is relatively weakly compact in c0 for every bounded operator T : X → c0. It is known that X has the Grothendieck property if and only if BX is a Grothendieck set. Of course the Grothendieck property implies both the positive and the weak Grothendieck properties. The lattice c of all real convergent sequences has the positive Grothendieck but it fails to have the weak Grothendieck property.
    [Show full text]
  • Boundedness and Surjectivity in Normed Spaces
    IJMMS 32:3 (2002) 149–165 PII. S0161171202011596 http://ijmms.hindawi.com © Hindawi Publishing Corp. BOUNDEDNESS AND SURJECTIVITY IN NORMED SPACES OLAV NYGAARD Received 24 January 2001 and in revised form 6 June 2001 We define the (w∗-) boundedness property and the (w∗-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w∗-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of two w∗-thick sets in X∗∗ and Y ∗ is a w∗-thick subset ∗ ∗ ∗ ∗ in L(X,Y ) and obtain as a consequence that the set w -expBK(l2) is w -thick. 2000 Mathematics Subject Classification: 46B20, 46B25, 28A33, 30H05. 1. Introduction. The following question is fundamental in the theory of linear op- erators: given two topological vector spaces U and V . Suppose we are given a linear continuous operator T : U → V and suppose we can show that the range of T contains a certain set A ⊂ V . Are there properties (S) such that the following is true: if A has the property (S), then T must be onto? In one dimension the following is of course true: suppose A ⊂ V contains one point different from the origin. Then TU ⊃ A implies TU = V . In arbitrary n-dimensional spaces V the theorem goes like this: suppose A ⊂ V contains n independent vectors. Then TU ⊃ A implies TU = V . Thus, the question is easy in finite-dimensional spaces.
    [Show full text]