Selection spectra and sharp perturbations of normal sequences
Wiesław Śliwa
Abstract
For a bounded sequence y = (y_n) in E* and 0 < p < infinity, let G_p(y) = {x in E : (y_n(x)) belongs to l_p}, and call y p-strongly normal when G_p(y) is dense in E. We study the spectra obtained by passing to subsequences and, for Schauder basic sequences, to normalized successive blocks. Their possible values are exactly empty set, (0,infinity), (1,infinity), [r,infinity), and (r,infinity) for r > 1, and the possible pairs are precisely the nested pairs. Every admissible pair has a normalized weakly null 1-unconditional basic realization, even in the fixed dual l_infinity(c) = l_1(c)*, and may be realized with empty spectrum for the original sequence.
We also determine sharp universal perturbation classes: l_p for p-strong normality at a fixed exponent, c_0 for the whole subsequence spectrum, and l_1 for the block spectrum. More generally, for every proper symmetric Banach sequence ideal A contained properly in c_0, membership of the error sequence in A is exactly the universal condition preserving density of the A-summability domain.
Finally, we extend the author's 2012 selection example from exponent one to every p > 0 and to every proper symmetric Banach sequence ideal. The good subsequences are exactly those whose index sets have uniformly bounded runs of consecutive integers; in the good case the detected quotient is isomorphic to c_0 and the whole selected sequence is weak-star Schauder basic.