An ordinal indexing on the space of strictly singular operators

Beanland, Kevin

doi:10.1007/s11856-011-0023-7

Cited by 5 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Denote the subclass of S ξ -strictly singular operators by S ξ -S. It is clear that S 0 -S coincides with the class of compact operators. These classes where defined in [3] and have been extensively studied [2,11,14,48]. In particular, while it is shown that (S ξ -S) ξ<ω 1 separably refines S. In general, S ξ -S need not satisfy the additive property of being an operator ideal [45].…”

Section: Weakly Compact Operators and Subclasses An Operatormentioning

confidence: 99%

Genericity and Universality for Operator Ideals

Beanland¹,

Causey²

2017

Preprint

Self Cite

View full text Add to dashboard Cite

A bounded linear operator U between Banach spaces is universal for the complement of some operator ideal J if it is a member of the complement and it factors through every element of the complement of J. In the first part of this paper, we produce new universal operators for the complements of several ideals and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use Descriptive Set Theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal J generic if anytime an operator A has the property that every operator in J factors through a restriction of A, then every operator between separable Banach spaces factors through a restriction of A. We prove that many of classical operator ideals are generic (i.e. strictly singular, weakly compact, Banach-Saks) and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W.B. Johnson which states that there is no universal operator for the complement of the ideal of completely continuous operators.

show abstract

Section: Weakly Compact Operators and Subclasses An Operatormentioning

confidence: 99%

Genericity and Universality for Operator Ideals

Beanland¹,

Causey²

2017

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this paper when we refer to a Borel subset of L(X, Y ) it is understood that this is with respect to the Borel σ-algebra generated by the strong operator topology. There are several papers in which L(X, Y ) is considered with this structure [4,5,6].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [1] they prove that each weakly compact operator on X is strictly singular. It is shown in [4] that when the strictly singular operators have codimension-one in L(X) they are a Borel subset. It follows that the set of weakly compact operators on X is a Borel subset of L(X).…”

Section: Applicationsmentioning

confidence: 99%

Uniformly factoring weakly compact operators

Beanland

Freeman

2014

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In particular, the map T → ̺(T ) is an ordinal rank 1 on the set SS(X, Y ) of all strictly singular operators from X to Y . It was further studied in [3,8,11,23].…”

Section: Introductionmentioning

confidence: 99%

On Strictly Singular Operators Between Separable Banach Spaces

Beanland

Dodos

2010

Mathematika

Self Cite

View full text Add to dashboard Cite

Abstract. Let X and Y be separable Banach spaces and denote by SS(X, Y ) the subset of L(X, Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set SS(X, Y ). Our main results are summarized as follows. Firstly, we define a new rank r S on SS(X, Y ). We show that r S is a co-analytic rank and that dominates the rank ̺ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math., 169 (2009), 221-250]. Secondly, for every 1 ≤ p < +∞ we construct a Banach space Yp with an unconditional basis such that SS(ℓp, Yp) is a co-analytic non-Borel subset of L(ℓp, Yp) yet every strictly singular operator T : ℓp → Yp satisfies ̺(T ) ≤ 2. This answers a question of Argyros.

show abstract

An ordinal indexing on the space of strictly singular operators

Cited by 5 publications

References 19 publications

Genericity and Universality for Operator Ideals

Genericity and Universality for Operator Ideals

Uniformly factoring weakly compact operators

On Strictly Singular Operators Between Separable Banach Spaces

Contact Info

Product

Resources

About