Uniform factorization for compact sets of weakly compact operators

Mikkor, Kristel; Oja, Eve

doi:10.4064/sm174-1-7

Cited by 4 publications

(3 citation statements)

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“…The idea of factoring all operators in a set through a single Banach space has been considered previously for compact operators and compact sets of weakly compact operators [3,18,26]. In particular, Johnson constructed a reflexive Banach space Z K such that if X and Y are Banach spaces and either X * or Y has the approximation property then every compact operator T : X → Y factors through Z K [20].…”

Section: Introductionmentioning

confidence: 99%

Uniformly factoring weakly compact operators

Beanland

Freeman

2014

Journal of Functional Analysis

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Abstract. Let X and Y be separable Banach spaces. Suppose Y either has a shrinking basis or Y is isomorphic to C(2 N ) and A is a subset of weakly compact operators from X to Y which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis Z such that every T ∈ A factors through Z. Likewise, we prove that if A ⊂ L(X, C(2 N )) is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space Z with separable dual such that every T ∈ A factors through Z. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary.

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Section: Introductionmentioning

confidence: 99%

Uniformly factoring weakly compact operators

Beanland

Freeman

2014

Journal of Functional Analysis

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“…Then by Theorem 3.3 there exists a regular Borel measure µ on B X * × B Y * with ϕ = µ so that Finally we consider a factorization of elements in K w * (X * , Y ). The following theorem is essentially contained in Aron, Lindström, Ruess, Ryan [1], and Mikkor, Oja [14]. But we use Proposition 3.1 to slightly simplify the existing proof.…”

Section: (D) T Xmentioning

confidence: 99%

ON SPACES OF WEAK^*TO WEAK CONTINUOUS COMPACT OPERATORS

Kim¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. This paper is concerned with the space K w * (X * , Y ) of weak * to weak continuous compact operators from the dual space X * of a Banach space X to a Banach space Y . We show that if X * or Y * has the Radon-Nikodým property, C is a convex subset of K w * (X * , Y ) with 0 ∈ C and T is a bounded linear operator from X * into Y , then T ∈ C τc if and only if T ∈ {S ∈ C : S ≤ T } τc , where τc is the topology of uniform convergence on each compact subset of X, moreover, if T ∈ K w * (X * , Y ), here C need not to contain 0, then T ∈ C τc if and only if T ∈ C in the topology of the operator norm. Some properties of K w * (X * , Y ) are presented.

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“…Johnson [18, Theorem 1] and Figiel [13,Proposition 3.1] showed that for any 1 ≤ p ≤ ∞ the spaces C p have the following compact factorisation property: given any Banach spaces X, Y and a compact operator T ∈ K(X, Y), there is a closed infinite-dimensional subspace W ⊂ C p as well as compact operators A 0 ∈ K(X, W), B 0 ∈ K(W, Y) so that T = B 0 A 0 . Following Aron et al [3] and [27], we consider the direct sum…”

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confidence: 99%

The Quotient Algebra of Compact-by-Approximable Operators on Banach Spaces Failing the Approximation Property

Tylli

Wirzenius²

2019

J. Aust. Math. Soc.

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We initiate a study of structural properties of the quotient algebra K(X)/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 K(Z)/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 non-separable space c 0 (Γ) into K(Z F J )/A(Z F J ), where Z F J is a universal compact factorisation space arising from the work of Johnson and Figiel.

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Uniform factorization for compact sets of weakly compact operators

Cited by 4 publications

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Uniformly factoring weakly compact operators

Uniformly factoring weakly compact operators

ON SPACES OF WEAK^*TO WEAK CONTINUOUS COMPACT OPERATORS

The Quotient Algebra of Compact-by-Approximable Operators on Banach Spaces Failing the Approximation Property

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Uniform factorization for compact sets of weakly compact operators

Cited by 4 publications

References 0 publications

Uniformly factoring weakly compact operators

Uniformly factoring weakly compact operators

ON SPACES OF WEAK*TO WEAK CONTINUOUS COMPACT OPERATORS

The Quotient Algebra of Compact-by-Approximable Operators on Banach Spaces Failing the Approximation Property

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ON SPACES OF WEAK^*TO WEAK CONTINUOUS COMPACT OPERATORS