A lower bound for the differences of powers of linear operators

Jarmo Malinen; Olavi Nevanlinna; Ville Turunen; Zhijian Yuan

doi:10.1007/s10114-005-0765-4

A lower bound for the differences of powers of linear operators

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Abstract

Let T be a bounded linear operator in a Banach space, with σ(T) = {1}. In 1983, Esterle–Berkani’ s conjecture was proposed for the decay of differences (I − T) T n as follows: Either

lim inf n →∞ (n + 1)∥ (I − T) T n ∥ ≥ 1/e

or T = I. We prove this claim and discuss some of its consequences.

Original language	English
Pages (from-to)	745-748
Journal	Acta Mathematica Sinica, English Series
Volume	23
Issue number	4
DOIs	https://doi.org/10.1007/s10114-005-0765-4
Publication status	Published - 2007
MoE publication type	A1 Journal article-refereed

Keywords

Esterle–Berkani’s conjecture
Quasi–nilpotent linear operator
Differences of powers
Decay

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10.1007/s10114-005-0765-4

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abstract = "Let T be a bounded linear operator in a Banach space, with σ(T) = \{1\}. In 1983, Esterle–Berkani{\textquoteright} s conjecture was proposed for the decay of differences (I − T) T n as follows: Either lim inf n →∞ (n + 1)∥ (I − T) T n ∥ ≥ 1/e or T = I. We prove this claim and discuss some of its consequences.",

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T1 - A lower bound for the differences of powers of linear operators

AU - Malinen, Jarmo

AU - Nevanlinna, Olavi

AU - Turunen, Ville

AU - Yuan, Zhijian

PY - 2007

Y1 - 2007

N2 - Let T be a bounded linear operator in a Banach space, with σ(T) = {1}. In 1983, Esterle–Berkani’ s conjecture was proposed for the decay of differences (I − T) T n as follows: Either lim inf n →∞ (n + 1)∥ (I − T) T n ∥ ≥ 1/e or T = I. We prove this claim and discuss some of its consequences.

AB - Let T be a bounded linear operator in a Banach space, with σ(T) = {1}. In 1983, Esterle–Berkani’ s conjecture was proposed for the decay of differences (I − T) T n as follows: Either lim inf n →∞ (n + 1)∥ (I − T) T n ∥ ≥ 1/e or T = I. We prove this claim and discuss some of its consequences.

KW - Esterle–Berkani’s conjecture

KW - Quasi–nilpotent linear operator

KW - Differences of powers

KW - Decay

U2 - 10.1007/s10114-005-0765-4

DO - 10.1007/s10114-005-0765-4

M3 - Article

SN - 1439-8516

VL - 23

SP - 745

EP - 748

JO - Acta Mathematica Sinica, English Series

JF - Acta Mathematica Sinica, English Series

IS - 4

ER -

A lower bound for the differences of powers of linear operators

Abstract

Keywords

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