Weak Sequential Completeness of 𝑲(X,Y)

2013 ◽  
Vol 56 (3) ◽  
pp. 503-509 ◽  
Author(s):  
Qingying Bu
Keyword(s):  
Banach Spaces ◽  
Compact Operator ◽  
Approximation Property ◽  
Compact Operators ◽  
Weakly Compact ◽  
Sequential Completeness ◽  
Weakly Compact Operator ◽  
Compact Approximation

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.

EQUIVALENCE OF SEMI-NORMS RELATED TO SUPER WEAKLY COMPACT OPERATORS

2021 ◽  
pp. 1-13
Author(s):  
KUN TU
Keyword(s):  
Banach Spaces ◽  
Quantitative Method ◽  
Approximation Property ◽  
Essential Norm ◽  
Compact Operators ◽  
Sequence Spaces ◽  
Weakly Compact ◽  
The Family ◽  
Compact Approximation ◽  
Weakly Compact Operators

Abstract We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

scholarly journals Ideals of compact operators

2004 ◽  
Vol 77 (1) ◽  
pp. 91-110 ◽  
Author(s):  
Åsvald Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Approximation Property ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Compact Approximation

AbstractWe give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

scholarly journals Uniformly factoring weakly compact operators and parametrised dualisation

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Antunes ◽  
K. Beanland ◽  
B. M. Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Subset ◽  
Compact Operators ◽  
Schauder Basis ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Weakly Compact Operator ◽  
Borel Map ◽  
Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

scholarly journals Dual functors and the Radon-Nikodym property in the category of Banach spaces

1979 ◽  
Vol 27 (4) ◽  
pp. 479-494 ◽  
Author(s):  
John Wick Pelletier
Keyword(s):  
Banach Spaces ◽  
Direct Limit ◽  
Integral Operators ◽  
Compact Operators ◽  
Property A ◽  
Weakly Compact ◽  
Nikodym Property ◽  
Weakly Compact Operators

AbstractThe notion of duality of functors is used to study and characterize spaces satisfying the Radon-Nikodym property. A theorem of equivalences concerning the Radon-Nikodym property is proved by categorical means; the classical Dunford-Pettis theorem is then deduced using an adjointness argument. The functorial properties of integral operators, compact operators, and weakly compact operators are discussed. It is shown that as an instance of Kan extension the weakly compact operators can be expressed as a certain direct limit of ordinary hom functors. Characterizations of spaces satisfying the Radon-Nikodym property are then given in terms of the agreement of dual functors of the functors mentioned above.

scholarly journals THE QUOTIENT ALGEBRA OF COMPACT-BY-APPROXIMABLE OPERATORS ON BANACH SPACES FAILING THE APPROXIMATION PROPERTY

2019 ◽  
pp. 1-23
Author(s):  
HANS-OLAV TYLLI ◽  
HENRIK WIRZENIUS
Keyword(s):  
Banach Spaces ◽  
Structural Properties ◽  
Approximation Property ◽  
Quotient Algebra ◽  
Isomorphic Embedding ◽  
Compact Approximation

We initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$ into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$ , where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$ into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$ , where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.

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