Some improvements of the Katznelson- Tzafriri theorem on Hilbert space

Seifert, David

doi:10.1090/proc/12323

Cited by 6 publications

(10 citation statements)

References 24 publications

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“…Zarrabi [89,Theorem 2.6] considered representations of semigroups in this class by contractions on Hilbert spaces, obtaining a more general version of (5.1) and a similar formula for C 0 -semigroups. The paper [79] extends Theorem 2.5 to bounded representations of these semigroups on Hilbert spaces, and it also extends Zarrabi's result (5.1) for contractive representations on Hilbert spaces.…”

Section: ] Bercovici Wrotesupporting

confidence: 71%

Some developments around the Katznelson–Tzafriri theorem

Batty

Seifert

2022

Acta Sci. Math. (Szeged)

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This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that $${\rm lim}_{n \rightarrow \infty} ||T^{n}(I-T)|| = 0$$ lim n → ∞ | | T n ( I - T ) | | = 0 if T is a power-bounded operator on a Banach space and $$\sigma (T)\cap \mathbb{T} \subseteq\{1\}$$ σ ( T ) ∩ T ⊆ { 1 } . Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.

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Section: ] Bercovici Wrotesupporting

confidence: 71%

Some developments around the Katznelson–Tzafriri theorem

Batty

Seifert

2022

Acta Sci. Math. (Szeged)

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“…Let X be a complex Banach space and let T ∈ B(X) be a powerbounded operator. Then Since its discovery in 1986, the Katznelson-Tzafriri theorem has attracted a considerable amount of interest, and this has lead to a number of extensions and improvements of the original result; see [12,Section 4] for an overview, and also [29], [39] and [41]. One aspect which so far has been studied only in special cases, however, is the rate at which decay takes place in (1.1); see for instance [13], [15], [35,Chapter 4], [36] and [37].…”

Section: Introductionmentioning

confidence: 99%

Rates of decay in the classical Katznelson-Tzafriri theorem

Seifert

2016

JAMA

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Given a power-bounded operator T , the theorem of Katznelson and Tzafriri states that T n (I −T ) → 0 as n → ∞ if and only if the spectrum σ(T ) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T ) ∩ T = {1}. The results obtained lead in particular to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(e iθ , T ) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.

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“…There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,5,7,9,11,14,15,16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…(1.1) have been obtained, see e.g. [1,3,4,5,6,7,9,11,14,15,16]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that sup n∈N T n < ∞, (1.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems

Minh¹,

Matsunaga²,

Huy³

et al. 2020

Preprint

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In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by n ν , where ν is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form ∆ α x(n) = T x(n) + y(n), n ∈ N, where 0 < α ≤ 1. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the αresolvent operator Sα satisfies sup n∈N Sα(n) /n ν < ∞ and the set of z 0 ∈ C such that (z − kα (z)T ) −1 exists, and together with kα (z), is holomorphic in a neighborhood of z 0 consists of at most 1, where kα (z) is the Z-transform of

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Some improvements of the Katznelson- Tzafriri theorem on Hilbert space

Cited by 6 publications

References 24 publications

Some developments around the Katznelson–Tzafriri theorem

Some developments around the Katznelson–Tzafriri theorem

Rates of decay in the classical Katznelson-Tzafriri theorem

A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems

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