Selection spectra and sharp perturbations of normal sequences

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. Th