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Journal articles on the topic '(order-)weakly compact operator'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Aqzzouz, Belmesnaoui, and Aziz Elbour. "Some characterizations of order weakly compact operator." Mathematica Bohemica 136, no. 1 (2011): 105–12. http://dx.doi.org/10.21136/mb.2011.141454.

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2

Aqzzouz, Belmesnaoui, and Jawad Hmichane. "Some results on order weakly compact operators." Mathematica Bohemica 134, no. 4 (2009): 359–67. http://dx.doi.org/10.21136/mb.2009.140668.

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3

AQZZOUZ, BELMESNAOUI, and JAWAD HMICHANE. "THE DUALITY PROBLEM FOR THE CLASS OF ORDER WEAKLY COMPACT OPERATORS." Glasgow Mathematical Journal 51, no. 1 (2009): 101–8. http://dx.doi.org/10.1017/s0017089508004576.

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AbstractWe study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

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4

Borwein, Jonathan M. "Generic differentiability of order-bounded convex oparators." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 28, no. 1 (1986): 22–29. http://dx.doi.org/10.1017/s0334270000005166.

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We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.

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5

Lhaimer, Driss, Khalid Bouras, and Mohammed Moussa. "On the class of order L-weakly and order M-weakly compact operators." Positivity 25, no. 4 (2021): 1569–78. http://dx.doi.org/10.1007/s11117-021-00829-2.

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6

AQZZOUZ, BELMESNAOUI, OTHMAN ABOUTAFAIL, and AZIZ ELBOUR. "On the weak compactness of the product of some operators." Creative Mathematics and Informatics 20, no. 2 (2011): 107–10. http://dx.doi.org/10.37193/cmi.2011.02.07.

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7

Elbour, Aziz, Nabil Machrafi, and Mohammed Moussa. "Weak Compactness of Almost Limited Operators." Journal of Function Spaces 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/263159.

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This paper is devoted to the relationship between almost limited operators and weakly compact operators. We show that ifFis aσ-Dedekind complete Banach lattice, then every almost limited operatorT:E→Fis weakly compact if and only ifEis reflexive or the norm ofFis order continuous. Also, we show that ifEis aσ-Dedekind complete Banach lattice, then the square of every positive almost limited operatorT:E→Eis weakly compact if and only if the norm ofEis order continuous.

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8

Akay, Barış, and Ömer Gök. "On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators." Journal of Function Spaces 2021 (May 20, 2021): 1–5. http://dx.doi.org/10.1155/2021/1755373.

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We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

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9

Nowak, Marian. "Yosida-Hewitt type decompositions for order-weakly compact operators." Bulletin of the Belgian Mathematical Society - Simon Stevin 18, no. 2 (2011): 259–69. http://dx.doi.org/10.36045/bbms/1307452076.

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10

Aqzzouz, Belmesnaoui, and Jawad H’Michane. "The b-Weak Compactness of Order Weakly Compact Operators." Complex Analysis and Operator Theory 7, no. 1 (2011): 3–8. http://dx.doi.org/10.1007/s11785-011-0138-1.

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11

Wolf, Elke. "WEAKLY COMPACT DIFFERENCES OF (WEIGHTED) COMPOSITION OPERATORS." Asian-European Journal of Mathematics 04, no. 04 (2011): 695–703. http://dx.doi.org/10.1142/s1793557111000575.

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12

Nowak, Marian. "Continuous linear operators on Orlicz-Bochner spaces." Open Mathematics 17, no. 1 (2019): 1147–55. http://dx.doi.org/10.1515/math-2019-0089.

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Abstract Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The r

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13

Lhaimer, Driss, Mohammed Moussa, and Khalid Bouras. "On the class of $\text{b}$-L-weakly and order M-weakly compact operators." Mathematica Bohemica 145, no. 3 (2019): 255–64. http://dx.doi.org/10.21136/mb.2019.0116-18.

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14

Nowak, Marian. "Order-weakly compact operators from vector-valued function spaces to Banach spaces." Proceedings of the American Mathematical Society 135, no. 09 (2007): 2803–10. http://dx.doi.org/10.1090/s0002-9939-07-08828-4.

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15

El Fahri, Kamal, and Jawad H’michane. "On the Product of Almost Dunford–Pettis and Order Weakly Compact Operators." Complex Analysis and Operator Theory 10, no. 3 (2015): 605–15. http://dx.doi.org/10.1007/s11785-015-0506-3.

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16

Di Pietro, Daniele A., Alexandre Ern, and Simon Lemaire. "An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators." Computational Methods in Applied Mathematics 14, no. 4 (2014): 461–72. http://dx.doi.org/10.1515/cmam-2014-0018.

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AbstractWe develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In t

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17

Gordin, V. A. "COMPACT FINITE-DIFFERENCE SCHEMES FOR WEAKLY NON-LINEAR PROBLEMS AND BOUNDARY CONDITIONS IMITATING CAUCHY PROBLEM." XXII workshop of the Council of nonlinear dynamics of the Russian Academy of Sciences 47, no. 1 (2019): 32–37. http://dx.doi.org/10.29006/1564-2291.jor-2019.47(1).9.

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Compact finite-difference schemes are well known and provide high accuracy order for differential equation with constant coefficients. Algorithms for constructing compact schemes of the 4-th order for boundary value problems with variable (smooth or jump) coefficient are developed. For the diffusion equations with a smooth variable coefficient and the Levin – Leontovich equation, compact finite-difference schemes are also constructed and their 4-th order is experimentally confirmed. The method of constructing compact schemes of the 4-th order can be generalized to partial differential equation

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18

Ayanbayev, Birzhan, and Nikos Katzourakis. "On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE." Vietnam Journal of Mathematics 49, no. 3 (2021): 815–29. http://dx.doi.org/10.1007/s10013-021-00515-6.

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AbstractIn this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the $L^{\infty }$ L ∞ minimisation problem whi

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19

Brattka, Vasco, and Guido Gherardi. "Weihrauch degrees, omniscience principles and weak computability." Journal of Symbolic Logic 76, no. 1 (2011): 143–76. http://dx.doi.org/10.2178/jsl/1294170993.

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AbstractIn this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The importance of Weihrauch degrees is based on the fact that m

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20

Dodds, Peter G., Theresa K. Dodds, and Ben De Pagter. "Weakly compact subsets of symmetric operator spaces." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 1 (1991): 169–82. http://dx.doi.org/10.1017/s0305004100070225.

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AbstractUnder natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

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21

Groenewegen, G., and A. van Rooij. "The modulus of a weakly compact operator." Mathematische Zeitschrift 195, no. 4 (1987): 473–80. http://dx.doi.org/10.1007/bf01166700.

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22

TURAN, Bahri, and Birol ALTIN. "The relation between b-weakly compact operator and KB-operator." TURKISH JOURNAL OF MATHEMATICS 43, no. 6 (2019): 2818–20. http://dx.doi.org/10.3906/mat-1908-11.

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23

Bu, Qingying. "Weak Sequential Completeness of 𝑲(X,Y)". Canadian Mathematical Bulletin 56, № 3 (2013): 503–9. http://dx.doi.org/10.4153/cmb-2011-202-9.

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AbstractFor Banach spaces X and Y, we show that if X* and Y are weakly sequentially complete and every weakly compact operator from X to Y is compact, then the space of all compact operators from X to Y is weakly sequentially complete. The converse is also true if, in addition, either X* or Y has the bounded compact approximation property.

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24

Albuquerque, R., S. Narison, F. Fanomezana, A. Rabemananjara, D. Rabetiarivony, and G. Randriamanatrika. "XYZ-like spectra from Laplace sum rule at N2LO in the chiral limit." International Journal of Modern Physics A 31, no. 36 (2016): 1650196. http://dx.doi.org/10.1142/s0217751x16501967.

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We present new compact integrated expressions of QCD spectral functions of heavy-light molecules and four-quark [Formula: see text]-like states at lowest order (LO) of perturbative (PT) QCD and up to [Formula: see text] condensates of the Operator Product Expansion (OPE). Then, by including up to next-to-next leading order (N2LO) PT QCD corrections, which we have estimated by assuming the factorization of the four-quark spectral functions, we improve previous LO results from QCD spectral sum rules (QSSR), on the [Formula: see text]-like masses and decay constants which suffer from the ill-defi

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25

Song, Xueli, and Jigen Peng. "On strong convex compactness property of spaces of nonlinear operators." Bulletin of the Australian Mathematical Society 74, no. 3 (2006): 411–18. http://dx.doi.org/10.1017/s0004972700040466.

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The strong convex compactness property is important for property persistence of operator semigroups under perturbations. It has been investigated in the linear setting. In this paper, we are concerned with the property in the nonlinear setting. We prove that the following spaces of (nonlinear) operators enjoy the strong convex compactness property: the space of compact operators, the space of completely continuous operators, the space of weakly compact operators, the space of conditionally weakly compact operators, the space of weakly completely continuous operators, the space of demicontinuou

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26

Aqzzouz, Belmesnaoui, and Khalid Bouras. "Some characterizations of weakly compact operator on Banach lattices." Czechoslovak Mathematical Journal 61, no. 4 (2011): 901–8. http://dx.doi.org/10.1007/s10587-011-0057-3.

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27

Chō, Muneo, and Tadasi Huruya. "A remark on the slice map problem." International Journal of Mathematics and Mathematical Sciences 17, no. 2 (1994): 401–4. http://dx.doi.org/10.1155/s0161171294000554.

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It is shown that there exist aσ-weakly closed operator algebraA˜, generated by finite rank operators and aσ-weakly closed operator algebraB˜generated by compact operators such that the Fubini productA˜⊗¯FB˜contains properlyA˜⊗¯B˜.

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28

Mathieu, Martin. "Properties of the Product of Two Derivations of a C*-Algebra." Canadian Mathematical Bulletin 32, no. 4 (1989): 490–97. http://dx.doi.org/10.4153/cmb-1989-072-4.

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29

Gretsky, Neil E., and Joseph M. Ostroy. "The compact range property and C0." Glasgow Mathematical Journal 28, no. 1 (1986): 113–14. http://dx.doi.org/10.1017/s0017089500006406.

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The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

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30

GALINDO, PABLO, and MIKAEL LINDSTRÖM. "WEAKLY COMPACT HOMOMORPHISMS BETWEEN SMALL ALGEBRAS OF ANALYTIC FUNCTIONS." Bulletin of the London Mathematical Society 33, no. 6 (2001): 715–26. http://dx.doi.org/10.1112/s0024609301008402.

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The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

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31

BEZHANISHVILI, GURAM, LEO ESAKIA, and DAVID GABELAIA. "THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE." Review of Symbolic Logic 3, no. 1 (2010): 26–40. http://dx.doi.org/10.1017/s1755020309990335.

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We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.

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32

YE, YUAN-LING. "Ruelle operator with weakly contractive iterated function systems." Ergodic Theory and Dynamical Systems 33, no. 4 (2012): 1265–90. http://dx.doi.org/10.1017/s0143385712000211.

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AbstractThe Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS $(X, \{w_j\}_{j=1}^m)$ and an associated family of positive continuous potential functions $\{p_j\}_{j=1}^m$, a triple system $(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ru

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33

Li, Li Mei, and Hong Luo. "Existence of Minimizer to the Ginzburg-Landau Free Energy of Ferromagnetic System." Applied Mechanics and Materials 444-445 (October 2013): 717–22. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.717.

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In this paper, we obtain the existence of minimizer to Ginzburg-Landau free energy of ferromagnetic system by coercivity and weakly lower semi-continuity of the free energy, where the weakly lower semi-continuity is derived from monotone operator condition and the Sobolev space compact imbedding theorem.

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34

Bonet, José, Paweł Dománski, and Mikael Lindström. "Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions." Canadian Mathematical Bulletin 42, no. 2 (1999): 139–48. http://dx.doi.org/10.4153/cmb-1999-016-x.

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AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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35

Albanese, Angela A., José Bonet, and Werner J. Ricker. "Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces." Abstract and Applied Analysis 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/179027.

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Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operatorTto the operator norm convergence of certain sequences of operators generated byT, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

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36

González, Manuel, and Antonio Martinón. "On operator ideals determined by sequences." Bulletin of the Australian Mathematical Society 44, no. 2 (1991): 285–95. http://dx.doi.org/10.1017/s0004972700029737.

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We associate with an operator ideal 𝒜 (in the sense of Pietsch) a class of bounded sequences S𝒜 by using the 𝒜-variation of Astala. If 𝒜 and B are operator ideals, and we define (𝒜, B) as the class of operators which map a sequence of S𝒜 into a sequence of SB, we obtain the following:Theorem. If Tn: X → Y is a sequence of operators and for every sequence (xn) ⊂ X in S𝒜 there exists p such that (Tpxn) belongs to SB, then Tm ∈ (𝒜, B) for some m.The compact operators, weakly compact operators and some other operator ideals can be represented as (𝒜, B). Hence several results of Tacon and other aut

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37

González, Manuel, and Joaquí M. Gutiérrez. "Polynomial Grothendieck properties." Glasgow Mathematical Journal 37, no. 2 (1995): 211–19. http://dx.doi.org/10.1017/s0017089500031116.

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AbstractA Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of

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38

Ben Amar, Afif, Mohamed Amine Cherif, and Maher Mnif. "Fixed-Point Theory on a Frechet Topological Vector Space." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/390720.

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We establish some versions of fixed-point theorem in a Frechet topological vector spaceE. The main result is that every mapA=BC(whereBis a continuous map andCis a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem forU-contractions and weakly compact mappings, while the second one, by assuming that the family{T(⋅,

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39

Nowak, Marian. "Yosida–Hewitt type decompositions for weakly compact operators and operator-valued measures." Journal of Mathematical Analysis and Applications 336, no. 1 (2007): 93–100. http://dx.doi.org/10.1016/j.jmaa.2007.02.057.

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40

Castillo, Jesús M. F., and Manuel González. "Properties (V) and (u) are not three-space properties." Glasgow Mathematical Journal 36, no. 3 (1994): 297–99. http://dx.doi.org/10.1017/s0017089500030895.

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In his fundamental papers [7,8], Pelczynski introduced properties (u), (V), and (V*) as tools as study the structure of Banach spaces. Let X be a Banach space. It is said that X has property (u) if, for every weak Cauchy sequence (xn) in X, there exists a weakly unconditionally Cauchy (wuC) series in X such that the sequence is weakly null. It is said that X has property (V) if, for every Banach space Z, every unconditionally converging operator from X into Z is weakly compact; equivalently, whenever K is a bounded subset of X* such that for every wuC series in X, then K is relatively weakly c

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41

Ricker, W. "Spectral operators and weakly compact homomorphisms in a class of Banach Spaces." Glasgow Mathematical Journal 28, no. 2 (1986): 215–22. http://dx.doi.org/10.1017/s0017089500006534.

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The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no

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42

Drewnowski, Lech. "Copies of l∞ in an operator space." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (1990): 523–26. http://dx.doi.org/10.1017/s0305004100069401.

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Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

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43

RUS, IOAN A. "Heuristic introduction to weakly Picard operator theory." Creative Mathematics and Informatics 23, no. 2 (2014): 243–52. http://dx.doi.org/10.37193/cmi.2014.02.06.

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In this paper we study the impact of weakly Picard operator theory, [see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191–219] on the following problem: what can we do in order to find conditions under which a given operator is a weakly Picard operator?

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44

Forrest, Brian E., Volker Runde, and Nico Spronk. "Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm." Canadian Journal of Mathematics 59, no. 5 (2007): 966–80. http://dx.doi.org/10.4153/cjm-2007-041-9.

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AbstractLet G be a locally compact group, and let Acb(G) denote the closure of A(G), the Fourier algebra of G, in the space of completely boundedmultipliers of A(G). If G is a weakly amenable, discrete group such that C*(G) is residually finite-dimensional, we show that Acb(G) is operator amenable. In particular, Acb() is operator amenable even though , the free group in two generators, is not an amenable group. Moreover, we show that if G is a discrete group such that Acb(G) is operator amenable, a closed ideal of A(G) is weakly completely complemented in A(G) if and only if it has an approxi

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45

TALPONEN, JARNO. "OPERATORS ON C0(L,X) WHOSE RANGE DOES NOT CONTAIN c0." Bulletin of the Australian Mathematical Society 77, no. 3 (2008): 515–20. http://dx.doi.org/10.1017/s0004972708000646.

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AbstractThis paper contains two results: (a) if $\mathrm {X}\neq \{0\}$ is a Banach space and (L,τ) is a nonempty locally compact Hausdorff space without isolated points, then each linear operator T:C0(L,X)→C0(L,X) whose range does not contain an isomorphic copy of c00 satisfies the Daugavet equality $\|\mathbf {I}+T\|=1+\|T\|$; (b) if Γ is a nonempty set and X and Y are Banach spaces such that X is reflexive and Y does not contain c0 isomorphically, then any continuous linear operator T:c0(Γ,X)→Y is weakly compact.

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46

Wright, J. D. Maitland. "HYPERMEASURE THEORY." Asian-European Journal of Mathematics 02, no. 03 (2009): 477–85. http://dx.doi.org/10.1142/s1793557109000406.

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Much of classical vector measure theory can be interpreted as the study of weakly compact operators on commutative function algebras. Non-commutative measure theory can be thought of as a similar study, where the domain algebras are replaced by non-commutative operator algebras. Hypermeasure theory is something beyond this, where we replace operator algebras by more general classes of Banach space.

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Dodds, P. G. "The range of an o-weakly compact mapping." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 39, no. 3 (1985): 391–99. http://dx.doi.org/10.1017/s144678870002615x.

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AbstractIt is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

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Leung, Denny H. "Banach spaces with property (w)." Glasgow Mathematical Journal 35, no. 2 (1993): 207–17. http://dx.doi.org/10.1017/s0017089500009769.

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A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type

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Bator, Elizabeth M., та Paul W. Lewis. "Properties (V) and (w V) on C(Ω X)". Mathematical Proceedings of the Cambridge Philosophical Society 117, № 3 (1995): 469–77. http://dx.doi.org/10.1017/s0305004100073308.

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A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakl

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Emmanuele, G. "Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E)." Glasgow Mathematical Journal 31, no. 2 (1989): 137–40. http://dx.doi.org/10.1017/s0017089500007655.

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Let K be a compact Hausdorff topological space and E be a Banach space not containing l1. Recently N. J. Kalton, E. Saab and P. Saab ([5]) obtained the results that under the above assumptions the usual space C(K, E) has the Dieudonné property; i.e. each weakly completely continuous operator on C(K, E) is weakly compact. They use topological results concerning multivalued mappings in their proof. In this short note we furnish a new and simpler proof of that result without using topological results but only well known theorems of Bourgain ([2]) and Talagrand ([8]) on weak compactness of sets of

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