scholarly journals The weakly compact reflection principle need not imply a high order of weak compactness

2019 ◽  
Vol 59 (1-2) ◽  
pp. 179-196
Author(s):  
Brent Cody ◽  
Hiroshi Sakai
Keyword(s):  
Weak Compactness ◽  
Reflection Principle ◽  
High Order ◽  
Weakly Compact

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

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.

scholarly journals PROOF THEORY OF WEAK COMPACTNESS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350003 ◽  
Author(s):  
TOSHIYASU ARAI
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

Weak compactness and square bracket partition relations

1972 ◽  
Vol 37 (4) ◽  
pp. 673-676 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Weak Compactness ◽  
Partition Relation ◽  
Tree Property ◽  
Weakly Compact ◽  
Partition Relations

Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for κ and α cardinals and n an integer the partition relation of Erdös, Hajnal and Rado [2] as follows:For every function F: [κ]n→ α (called a partition of [κ]n, the n-element subsets of κ, into α pieces), there exists a set C⊆ κ (called homogeneous for F) such that card C = κ and F″[C]n≠ α, i.e. some element of the range is omitted when F is restricted to the n-element subsets of C. It is the simplest (nontrivial) of these relations, i.e. , that is the well-known equivalent of weak compactness.1Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript—e.g., —and (b) trying to weaken the relation by increasing the subscript—e.g., . As it turns out, the strengthening to is only illusory for using the equivalence of to the tree property one quickly sees that implies (and so is equivalent to) for every n. Thus is the strongest of these partition relations. The second question seems much more difficult.

scholarly journals THE DUALITY PROBLEM FOR THE CLASS OF ORDER WEAKLY COMPACT OPERATORS

2009 ◽  
Vol 51 (1) ◽  
pp. 101-108 ◽  
Author(s):  
BELMESNAOUI AQZZOUZ ◽  
JAWAD HMICHANE
Keyword(s):  
Necessary Conditions ◽  
Compact Operators ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Weakly Compact Operators ◽  
Sufficient And Necessary Conditions ◽  
Duality Problem

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.

Weak compactness of direct sums in locally convex cones

2018 ◽  
Vol 55 (4) ◽  
pp. 487-497
Author(s):  
Mohammad Reza Motallebi
Keyword(s):  
Weak Compactness ◽  
Convex Cones ◽  
Weakly Compact ◽  
Direct Sums ◽  
Direct Sum ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Compact Subsets

We discuss the weakly compact subsets of direct sum cones for the upper, lower and symmetric topologies and investigate the X-topologies of the weak upper, lower and sym-metric compact subsets of direct sum cones on product cones.

scholarly journals Remarks on Weak Compactness in L1(μ,X)

1977 ◽  
Vol 18 (1) ◽  
pp. 87-91 ◽  
Author(s):  
J. Diestel
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Factorization Method ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Potential Applicability ◽  
Image Position

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed byThe problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

Weak compactness of almost L-weakly and almost M-weakly compact operators

2020 ◽  
pp. 1-10
Author(s):  
Farid Afkir ◽  
Khalid Bouras ◽  
Aziz Elbour ◽  
Safae El Filali
Keyword(s):  
Compact Operators ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Weakly Compact Operators

Weakly Compact Sets in Orlicz Spaces

1962 ◽  
Vol 14 ◽  
pp. 170-176 ◽  
Author(s):  
Tsuyoshi Andô
Keyword(s):  
Orlicz Space ◽  
Orlicz Spaces ◽  
Characterization Theorem ◽  
Weak Compactness ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Köthe Spaces ◽  
Continuous Seminorms ◽  
Weakly Compact Sets

The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.

Sign in / Sign up

Export Citation Format

Share Document