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Power‐Bounded Operators and Related Norm Estimates
Author(s) -
Kalton N.,
Montgomery-Smith S.,
Oleszkiewicz K.,
Tomilov Y.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005514
Subject(s) - bounded function , mathematics , spectral radius , norm (philosophy) , operator (biology) , bounded operator , operator norm , power (physics) , combinatorics , constant (computer programming) , discrete mathematics , pure mathematics , mathematical analysis , physics , eigenvalues and eigenvectors , computer science , biochemistry , chemistry , repressor , quantum mechanics , political science , transcription factor , law , gene , programming language
It is considered whether L = lim sup n → ∞ n ‖T n + 1 − T n ‖< ∞ implies that the operator T is power‐bounded. It is shown that this is so if L <1/e, but it does not necessarily hold if L =1/e. As part of the methods, a result of Esterle is improved, showing that if σ( T )={1} and T {≠} I , then lim inf n → ∞ n ‖T n + 1 − T n ‖⩾ 1 / e . The constant 1/e is sharp. Finally, a way to create many generalizations of Esterle's result is described, and also many conditions are given on an operator which imply that its norm is equal to its spectral radius.