Kevin Beanland

An extremely non-homogeneous weak Hilbert space

We construct a weak Hilbert Banach space such that for every block subspace Y every bounded linear operator on Y is of the form D + S, where S is a strictly singular operator and D is a diagonal operator. We show that this yields a weak Hilbert space whose block subspaces are not isomorphic to any of their proper subspaces.

Descriptive Set Theoretic Methods Applied to Strictly Singular and Strictly Cosingular Operators

The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω 1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.

On strictly singular operators between separable Banach spaces

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 it dominates the rank ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [ Israel J. Math. 169 (2009), 221–250]. Secondly, for every 1≤ p ∞ , we construct a Banach space Y p with an unconditional basis such that 𝒮𝒮( l p , Y p ) is a co-analytic non-Borel subset of ℒ( l p , Y p ) yet every strictly singular operator T : l p → Y p satisfies ϱ ( T )≤2. This answers a question of Argyros.

Modifications of Thomae's Function and Differentiability

As we shall see later, Thomae’s function is not differentiable on the irrationals. In this note, we address whether there is a modification of Thomae’s function which is differentiable on a subset of the irrationals. In Section 2, we prove that Thomae’s function is not differentiable on the irrationals and define modified versions of Thomae’s function. In Section 3, we show that for each of our modifications there is a dense subset of irrationals on which, quite surprisingly, the function is not differentiable. Finally, in Section 4, we show that the measure of irrationality of a given number determines which modifications of Thomae’s function are differentiable at that number.

THE STABILIZED SET OF p'S IN KRIVINE'S THEOREM CAN BE DISCONNECTED

Abstract For any closed subset F of [ 1 , ∞ ] which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space X with a 1-unconditional basis is constructed so that in each block subspace Y of X, l p is finitely block represented in Y if and only if p ∈ F . In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace Y of X there is a dense subset G of F such that the spreading models admitted by Y are exactly the l p for p ∈ G .

Operators on the ∞-spaces of Bourgain and Delbaen

Abstract In 1980, J. Bourgain and F. Delbaen constructed two classes and of ∞-spaces each exhibiting many surprising properties. In particular, the members of each possess the Schur property and the members of are somewhat reflexive. In this paper, for a Banach space in either of these classes we define a bounded shift-type operator which is an isometry when restricted to a certain hyperplane.