Weakly null sequences with upper \(\ell_ p\)-estimates. (English) Zbl 0759.46013

Functional analysis, Proc. Semin., Austin/TX (USA) 1987-89, Lect. Notes Math. 1470, 85-107 (1991).
[For the entire collection see Zbl 0731.00013.]
The authors define the condition \((S_ p)\). A Banach space \(X\) has property \((S_ p)\) if every weakly null sequence \((x_ n)\) has a subsequence \((y_ n)\) such that \(\|\alpha_ 1 y_ 1+\alpha_ 2 y_ 2+\dots\|\leq C<\infty\) for all real sequence \((\alpha_ n)\) with \((| \alpha_ 1|^ p+| \alpha_ 2|^ p+\dots)^{1\over p}\leq 1\), where \(1<p<\infty\). The purpose of the present paper is to give those new properties which are equivalent or which are strictly connected to the condition \((S_ p)\). The paper contains some examples of Banach spaces which satisfy the considered properties and the proofs of the main theorem and some version of this theorem.
Reviewer: A.Waszak (Poznań)


46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces


Zbl 0731.00013