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A Sequentially Unconditional Banach Space with Few Operators
Author(s) -
Argyros S. A.,
Manoussakis A.
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611505015388
Subject(s) - mathematics , banach space , schauder basis , subsequence , infinite dimensional vector function , injective function , function space , pure mathematics , sequence (biology) , discrete mathematics , eberlein–šmulian theorem , reflexive space , lp space , interpolation space , mathematical analysis , functional analysis , gene , biology , bounded function , genetics , biochemistry , chemistry
A Banach space X is said to be sequentially unconditional if every Schauder basic sequence has an unconditional subsequence. We provide an example of a reflexive Banach space X isu which is sequentially unconditional and such that every T ∈ L( X isu ) is a strictly singular perturbation of a multiple of the identity. The space X isu belongs to the class of Banach spaces with saturated and conditional structure. Constructing X isu we follow the general scheme invented by W. T. Gowers and B. Maurey in their celebrated construction of a Hereditarily Indecomposable Banach space. The new ingredient concerns the definition of the special (conditional) functionals. For this we use a non‐injective coding function σ in conjunction with a Baire‐like property of a sequence of countable trees stated and proved in the present paper. It is worth mentioning that X isu admits a richer unconditional structure than classical spaces like L 1 (0,1) and it remains unsettled whether X isu is a subspace of a Banach space with an unconditional basis. 2000 Mathematics Subject Classification 46B20, 46B45 (primary), 05D99, 47L05 (secondary).