Remarks on compact approximation in Banach spaces

1974 ◽  
Vol 15 (1) ◽  
pp. 143-146
Author(s):  
Yu. N. Vladimirskii
Keyword(s):  
Banach Spaces ◽  
Compact Approximation

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.

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 Weakly compact approximation in Banach spaces

2004 ◽  
Vol 357 (3) ◽  
pp. 1125-1159 ◽  
Author(s):  
Edward Odell ◽  
Hans-Olav Tylli
Keyword(s):  
Banach Spaces ◽  
Weakly Compact ◽  
Compact Approximation

On commuting approximation properties of Banach spaces

2009 ◽  
Vol 139 (3) ◽  
pp. 551-565 ◽  
Author(s):  
Eve Oja ◽  
Indrek Zolk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Canonical Projection ◽  
Compact Set ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Image Position ◽  
Approximation Of The Identity

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequalityand 1 ≤ λ < max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.

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