Selection spectra and sharp perturbations of normal sequences

Abstract

For a bounded sequence $y=(y_n)\subset E^*$ and $0<p<\infty$, let $G_p(y)=\{x\in E:(y_n(x))\in\ell_p\},$ 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 $\varnothing,\quad (0,\infty),\quad (1,\infty),\quad [r,\infty),\quad (r,\infty)\qquad (r>1),$ 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 $\ell_\infty(\mathfrak c)=\ell_1(\mathfrak c)^*$, and may be realized with empty spectrum for the original sequence. We also determine sharp universal perturbation classes: $\ell_p$ for $p$-strong normality at a fixed exponent, $c_0$ for the whole subsequence spectrum, and $\ell_1$ for the block spectrum. More generally, for every proper symmetric Banach sequence ideal $A\subsetneq c_0$, 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.