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An extension of the Vu–Sine theorem and compact-supercyclicity

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
332
Issue
2
Identifiers
DOI: 10.1016/j.jmaa.2006.11.021
Keywords
  • [Formula Omitted]-Semigroup
  • Power Bounded Operator
  • Supercyclicity
  • Almost-Periodic Representation

Abstract

Abstract If ( T t ) t ⩾ 0 is a bounded C 0 -semigroup in a Banach space X and there exists a compact subset K ⊆ X such that lim inf t → ∞ ρ ( T t x , K ) = 0 ( ∀ x ∈ X , ‖ x ‖ ⩽ 1 ) , then there exists a finite-dimensional subspace L ⊆ X such that lim t → ∞ ρ ( T t x , L ) = 0 ( ∀ x ∈ X ) . If T : X → X ( X is real or complex) is supercyclic and ( ‖ T n ‖ ) n is bounded then ( T n x ) n vanishes for every x ∈ X . We define the “compact-supercyclicity.” If dim X = ∞ then X has no compact-supercyclic isometries.

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