Quasi-constricted linear operators on Banach spaces

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Emel'yanov E., Wolff M.

STUDIA MATHEMATICA, vol.144, no.2, pp.169-179, 2001 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 144 Issue: 2
  • Publication Date: 2001
  • Doi Number: 10.4064/sm144-2-5
  • Journal Name: STUDIA MATHEMATICA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.169-179
  • Middle East Technical University Affiliated: No

Abstract

Let X be a Banach space over C. The bounded linear operator T on X is called quasi-constricted if the subspace X-0 := {x epsilon X : lim(n --> infinity) parallel toT(n)x parallel to = 0} is closed and has finite codimension. We show that a power bounded linear operator T epsilon L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness chi parallel to (.)parallel to (1) (A) < 1 for some equivalent norm parallel to (.)parallel to (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 <()over bar>T is mean ergodic for all lambda in the peripheral spectrum sigma (pi)(T) of T is constricted and power bounded, and hence has a compact attractor.