Examples of hereditarilyl¹Banach spaces failing the Schur property

Abstract

A class of separable Banach sequence spaces is constructed. A member X of this class (i) is a hereditarily I 1 dual space which fails the Schur property, and (ii) is of codimension one in its first Baire class. A consequence of (ii) is that X is not isomorphic to the square of any Banach space Y.Introduction. In this paper we introduce and study a new class of Banach sequence spaces, the X a spaces. The definition of the norm in a particular X a space depends on the action of special sequences of intervals of integers on a vector x = (t l9 t 2 , -) (as in the definition of the James space / [6]) in conjunction with a fixed sequence of weighting factors (as in the Lorentz sequence spaces [7].) Let X denote a specific X a space, and let (e t ) denote the sequence of usual unit vectors in X (i.e. e t (j) = 8 tJ for integers / and j). Our main result is the following:

A hereditarily ℓ₁ subspace of 𝐿₁ without the Schur property

“…It is well known that 1 has the Schur property. As was shown by J. Bourgain and H. P. Rosenthal in [3], there exists a subspace of L 1 having the Schur property, but which does not embed in 1 . The first example of a hereditarily 1 Banach space without the Schur property was constructed by J. Bourgain in [2].…”

“…The first example of a hereditarily 1 Banach space without the Schur property was constructed by J. Bourgain in [2]. Then P. Azimi and J. N. Hagler in [1] constructed a class of such spaces and investigated their further properties. We construct a class of subspaces of L 1 with the same properties.…”

A class of Banach sequence spaces analogous to the space of Popov

“…Among other interesting properties it does not possess the Schur property [2]. Then these spaces were extended to a new class of hereditarily l p Banach spaces, X α,p [1].…”

“…in this norm. For a good information concerning these spaces, we refer to [1] and [2]. Now we go through the construction of the spaces X p analogous to the space of Popov.…”

Further properties of Azimi-Hagler banach spaces

“…These spaces are denoted by X α,p . In [3] classes of spaces containing hereditarily ℓ 1 which fail the Schur property were constructed and studied. In [1] classes of X α,p Banach spaces were constructed which are hereditarily complementably ℓ p .…”