A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Studia Mathematica (1996)

• Volume: 121, Issue: 2, page 167-183
• ISSN: 0039-3223

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Abstract

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Let T be a semigroup of linear contractions on a Banach space X, and let ${X}_{s}\left(T\right)=x\in X:li{m}_{s\to \infty }\parallel T\left(s\right)x\parallel =0$. Then ${X}_{s}\left(T\right)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then ${X}_{s}\left(T\right)$ is the annihilator of the unitary eigenvectors of T*, and $li{m}_{s}\parallel T\left(s\right)x\parallel =inf\parallel x-y\parallel :y\in {X}_{s}\left(T\right)$ for each x in X.

How to cite

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Batty, Charles, Brzeźniak, Zdzisław, and Greenfield, David. "A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum." Studia Mathematica 121.2 (1996): 167-183. <http://eudml.org/doc/216349>.

@article{Batty1996,
abstract = {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.},
author = {Batty, Charles, Brzeźniak, Zdzisław, Greenfield, David},
journal = {Studia Mathematica},
keywords = {contraction semigroup; unitary spectrum; unitary eigenvector trajectory; asymptotic stability; trivially asymptotically stable; countable; spectral synthesis; semigroup of linear contractions; annihilator; bounded trajectories},
language = {eng},
number = {2},
pages = {167-183},
title = {A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum},
url = {http://eudml.org/doc/216349},
volume = {121},
year = {1996},
}

TY - JOUR
AU - Batty, Charles
AU - Brzeźniak, Zdzisław
AU - Greenfield, David
TI - A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum
JO - Studia Mathematica
PY - 1996
VL - 121
IS - 2
SP - 167
EP - 183
AB - 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.
LA - eng
KW - contraction semigroup; unitary spectrum; unitary eigenvector trajectory; asymptotic stability; trivially asymptotically stable; countable; spectral synthesis; semigroup of linear contractions; annihilator; bounded trajectories
UR - http://eudml.org/doc/216349
ER -

References

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