Selection spectra and sharp perturbations of normal sequences
Abstract
For a bounded sequence y=(yn)⊂ E* and p>0, put Gp(y)={x∈ E:(yn(x))∈ℓp} and call y p-strongly normal when Gp(y) is dense in E. We study the density spectra of the original sequence, its subsequences, and its normalized successive blocks. The possible subsequence and block spectra are exactly ∅, (0,∞), (1,∞), [r,∞), and (r,∞) for r>1, and their possible pairs are precisely the nested pairs. Every such pair has a normalized weakly null $1$-unconditional basic realization in ℓ1(c)*, even with empty original-sequence spectrum. Without a basicness requirement, the possible original-sequence spectra are all upper subsets of (0,∞); each is realized in a separable Hilbert space. We determine the exact universal perturbation classes: ℓp for a fixed original-sequence exponent, c0 for the subsequence spectrum, and ℓ1 for the block spectrum. The ideal-valued version holds for every proper solid permutation-invariant linear ideal c00⊂ Asubsetneq c0, without a norm or completeness assumption. For the author's 2012 selection model, the bounded-run criterion extends to all proper symmetric quasi-Banach sequence ideals. In the unbounded-run case, a single norm-one projection annihilates the summability domains of any family of fewer than b such ideals; the quotient by their closed linear span contains a $1$-complemented isometric copy of ℓ∞. Such copies also occur in the left jump at every q>1 and in the right jump at every q≥1 of the scale of closed summability domains. Thus the common density criterion coexists with two-sided complemented discontinuities.
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BibTeX
@misc{sliwa2026selection,
title = {{Selection spectra and sharp perturbations of normal sequences}},
author = {Wiesław Śliwa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38853},
eprint = {2610.38853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38853},
}
RIS
TY - UNPB
TI - Selection spectra and sharp perturbations of normal sequences
AU - Wiesław Śliwa
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38853
UR - https://arxiv.org/abs/2610.38853
AB - For a bounded sequence y=(y_n)⊂ E^* and p>0, put G_p(y)={x∈ E:(y_n(x))∈ℓ_p} and call y p-strongly normal when G_p(y) is dense in E. We study the density spectra of the original sequence, its subsequences, and its normalized successive blocks. The possible subsequence and block spectra are exactly ∅, (0,∞), (1,∞), [r,∞), and (r,∞) for r>1, and their possible pairs are precisely the nested pairs. Every such pair has a normalized weakly null $1$-unconditional basic realization in ℓ_1(c)^*, even with empty original-sequence spectrum. Without a basicness requirement, the possible original-sequence spectra are all upper subsets of (0,∞); each is realized in a separable Hilbert space. We determine the exact universal perturbation classes: ℓ_p for a fixed original-sequence exponent, c_0 for the subsequence spectrum, and ℓ_1 for the block spectrum. The ideal-valued version holds for every proper solid permutation-invariant linear ideal c_00⊂ Asubsetneq c_0, without a norm or completeness assumption. For the author's 2012 selection model, the bounded-run criterion extends to all proper symmetric quasi-Banach sequence ideals. In the unbounded-run case, a single norm-one projection annihilates the summability domains of any family of fewer than b such ideals; the quotient by their closed linear span contains a $1$-complemented isometric copy of ℓ_∞. Such copies also occur in the left jump at every q>1 and in the right jump at every q≥1 of the scale of closed summability domains. Thus the common density criterion coexists with two-sided complemented discontinuities.
ER -