A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Volume 121 / 1996

Charles J. K. Batty, Zdzisław Brzeźniak, David D. Greenfield Studia Mathematica 121 (1996), 167-183 DOI: 10.4064/sm_1996_121_2_1_167_183


Let T be a semigroup of linear contractions on a Banach space X, and let $X_{s}(T) = {x ∈ X : lim_{s→∞} ∥T(s)x∥ = 0}$. Then $X_{s}(T)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_{s}(T)$ is the annihilator of the unitary eigenvectors of T*, and $lim_{s} ∥T(s)x∥ = inf{∥x-y∥ : y ∈ X_{s}(T)}$ for each x in X.


  • Charles J. K. Batty
  • Zdzisław Brzeźniak
  • David D. Greenfield

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