2011/01/01 by Spiros A. Argyros, Richard Haydon · 2 citations
Mathematics · #Advanced Banach Space Theory #Holomorphic and Operator Theory #Nonlinear Differential Equations Analysis #Mathematics #Indecomposable module #Banach space #Scalar (mathematics) #Bounded function #Pure mathematics #Space (punctuation) #Operator space #Bounded operator #Dual space #Discrete mathematics #Mathematical analysis #Finite-rank operator #Geometry
paper · pdf · doi:10.1007/s11511-011-0058-y
openalex publication_date 2011/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
s. a. argyros and r. g. haydon result to say that an HI predual of ℓ 1 necessarily has the scalar-plus-compact property.We use, in an essential way, the specific structure of the BD construction, which embeds into our space some very explicit finite-dimensional ℓ ∞ -spaces.As well as the (now) classical machinery of HI constructions-a space of Schlumprecht type (cf.[40]), Maurey-Rosenthal coding (cf.[35]) and rapidly increasing sequences based on ℓ 1 -averages-we add the possibility of splitting an arbitrary vector into pieces of comparable norm, while staying in one of these ℓ n ∞ 's.This allows us to introduce two additional classes of rapidly increasing sequences, and these in turn lead to the stronger result about operators. AcknowledgmentsMuch of the research presented in this paper was carried out during the second author's three visits to Athens in 2007 and 2008.He offers his thanks to the National Technical University for the support that made these visits possible and to all members of the NTU Analysis group for providing an outstanding research environment.Both authors would especially like to thank T. Raikoftsalis for many stimulating discussions.We are also very grateful to Dr. A. Tolias whose careful reading of an earlier version of the paper led to the correction of a (reprehensibly large) number of minor errors.Finally, we acknowledge with gratitude the hard work of the referee, who made valuable suggestions about the organization and structure of the paper, as well as identifying further errors and obscurities. Background NotationWe use standard notation: if A is any set, ℓ ∞ (A) is the space of all bounded (real-valued) functions on A, equipped with the supremum norm • ∞ and ℓ 1 (A) is the space of all absolutely summable functions on A, equipped with the norm x 1 = a∈A |x(a)|.The support of a function x is the set of all a such that x(a) =0; c 00 (A) is the space of functions of finite support.We shall write ℓ p for the space ℓ p (N), where N is the set 1, 2, 3, ... of positive integers, and ℓ n p for ℓ p (1, 2, ..., n).Even when we are dealing with these sequence spaces we shall use function notation x(m), rather than subscript notation, for the mth coordinate of the vector x.When x and y are in c 00 (A) (and more generally when the sum exists) we shall write y, x for a∈A x(a)y(a).If we are thinking of y as a functional acting on x (rather than vice versa) we shall usually choose a notation involving a star, denoting y by f * , or something of this kind.In particular, e a and e * a are two notations for the same unit vector s.a. argyros and r.