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Constructive Approximation

, Volume 1, Issue 1, pp 217–229 | Cite as

Asymptotic centers and best compact approximation of operators intoC(K)

  • Yoav Benyamini
Article

Abstract

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX*. IfK is just one convergent sequence, the condition is that everyω*-convergent sequence inX* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω2) and from ℓ1 intoL1 without best compact approximation. We also construct spacesX1,X2, isomorphic to a Hilbert space, and operatorsT1,∶X1C(ω2),T2∶ℓ1X2 without best compact approximations.

AMS classification

41A50 41A65 46B20 

Key words and phrases

Best approximations Linear operators Asymptotic centers 

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Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • Yoav Benyamini
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of TexasAustinUSA
  2. 2.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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