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We prove that the normalized sequence $$\big ((p\lambda _n+1)^{1/p}t^{\lambda _n})\big )_{n\in \mathbb {Z}}$$((pλn+1)1/ptλn))n∈Z in $$L^p([0,1])$$Lp([0,1]), up to some truncation, is asymptotically isometric to the canonical basis of $$\ell ^p$$ℓp if… Click to show full abstract
We prove that the normalized sequence $$\big ((p\lambda _n+1)^{1/p}t^{\lambda _n})\big )_{n\in \mathbb {Z}}$$((pλn+1)1/ptλn))n∈Z in $$L^p([0,1])$$Lp([0,1]), up to some truncation, is asymptotically isometric to the canonical basis of $$\ell ^p$$ℓp if and only if it is almost isometric if and only if $$(\lambda _n+1/p)_n$$(λn+1/p)n (resp. $$(\lambda _n)_n$$(λn)n) is a super-lacunary sequence. This extends recent results of the same authors. Similar results occur in C([0, 1]). As a particular application, we get that all the (strict) s-numbers of the classical Cesàro operator on $$L^p$$Lp are equal to $$p'$$p′ when $$p\in (1,+\infty )$$p∈(1,+∞).
Journal Title: Integral Equations and Operator Theory
Year Published: 2019
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