Weakly null sequences with upper lp-estimates

Weakly Null Sequences with Upper Estimates

Abstract. We prove that if (vi) is a normalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by (vi), then there exists a uniform constant C ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by (vi). This extends a result of Knaust and Odell, who proved this for the ca...

Weakly null sequences with upper `p-estimates

Weakly Null Sequences in L1

In [MR], the second author and H. P. Rosenthal constructed the first examples of weakly null normalized sequences which do not have any unconditionally basic subsequences. Much more recently, the second author and W. T. Gowers [GM] constructed infinite dimensional Banach spaces which do not contain any unconditionally basic sequences. These later examples are of a different character than the e...

On the Convergence in Mean of Martingale Difference Sequences

In [6] Freniche proved that any weakly null martingale difference sequence in L1[0, 1] has arithmetic means that converge in norm to 0. We show any weakly null martingale difference sequence in an Orlicz space whose N-function belongs to ∇3 has arithmetic means that converge in norm to 0. Then based on a theorem in Stout [13][Theorem 3.3.9 (i) and (iii)], we give necessary and sufficient condit...

Double-null operators and the investigation of Birkhoff's theorem on discrete lp spaces

Doubly stochastic matrices play a fundamental role in the theory of majorization. Birkhoff's theorem explains the relation between $ntimes n$ doubly stochastic matrices and permutations. In this paper, we first introduce double-null  operators and we will find some important properties of them. Then with the help of double-null operators, we investigate Birkhoff's theorem for descreate $l^p$ sp...