Measure of weak noncompactness under complex interpolation

Volume 147 / 2001

Andrzej Kryczka, Stanisław Prus Studia Mathematica 147 (2001), 89-102 MSC: 46B70, 46M35. DOI: 10.4064/sm147-1-7

Abstract

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón's complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if $T:A_{0}\rightarrow B_{0}$ or $T:A_{1}\rightarrow B_{1}$ is weakly compact, then so is $T:A_{[\theta ]}\rightarrow B_{[\theta ]}$ for all $0<\theta <1$, where $A_{[\theta ]}$ and $B_{[\theta ]}$ are interpolation spaces with respect to the pairs $(A_{0},A_{1})$ and $(B_{0},B_{1})$. Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

Authors

  • Andrzej KryczkaInstitute of Mathematics
    Maria Curie-Skłodowska University
    20-031 Lublin, Poland
    e-mail
  • Stanisław PrusInstitute of Mathematics
    Maria Curie-Skłodowska University
    20-031 Lublin, Poland
    e-mail

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