Symmetric block bases of sequences with large average growth

Abstract

We show that if 0<e≦1, 1≦p<2 andx 1, …,x n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ l n ei x l‖≧n 1/p, then one can find a block basisy 1, …,y m ofx 1, …,x n which is (1+e)-symmetric and has cardinality at leastγn 2/p-1(logn)−1, where γ depends on e only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex 1, …,x n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n 2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn 2/p-1(logn)4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.