Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel'yanov, and Manfred P. H. Wolff. "Quasi-constricted linear operators on Banach spaces." Studia Mathematica 144.2 (2001): 169-179. <http://eudml.org/doc/285196>.

@article{EduardYu2001,
abstract = {Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀: = \{x ∈ X: lim_\{n→ ∞\} ||Tⁿx|| = 0\}$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_\{||·||₁\}(A) < 1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove that every quasi-constricted operator T such that λ̅T is mean ergodic for all λ in the peripheral spectrum $σ_\{π\}(T)$ of T is constricted and power bounded, and hence has a compact attractor.},
author = {Eduard Yu. Emel'yanov, Manfred P. H. Wolff},
journal = {Studia Mathematica},
keywords = {quasi-constricted operators; attractor; measure of noncompactness; power bounded operator; mean ergodic operator; spectral radius},
language = {eng},
number = {2},
pages = {169-179},
title = {Quasi-constricted linear operators on Banach spaces},
url = {http://eudml.org/doc/285196},
volume = {144},
year = {2001},
}

TY - JOUR
AU - Eduard Yu. Emel'yanov
AU - Manfred P. H. Wolff
TI - Quasi-constricted linear operators on Banach spaces
JO - Studia Mathematica
PY - 2001
VL - 144
IS - 2
SP - 169
EP - 179
AB - Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀: = {x ∈ X: lim_{n→ ∞} ||Tⁿx|| = 0}$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_{||·||₁}(A) < 1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove that every quasi-constricted operator T such that λ̅T is mean ergodic for all λ in the peripheral spectrum $σ_{π}(T)$ of T is constricted and power bounded, and hence has a compact attractor.
LA - eng
KW - quasi-constricted operators; attractor; measure of noncompactness; power bounded operator; mean ergodic operator; spectral radius
UR - http://eudml.org/doc/285196
ER -