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Measure of weak noncompactness and real interpolation of operators

Published online by Cambridge University Press:  17 April 2009

Andrzej Kryczka ,
Stanislaw Prus  and
Mariusz Szczepanik
Andrzej Kryczka
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: akryczka@golem.umcs.lublin.plbsprus@golem.umcs.lublin.plszczepan@golem.umcs.lublin.pl
Stanislaw Prus
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: akryczka@golem.umcs.lublin.plbsprus@golem.umcs.lublin.plszczepan@golem.umcs.lublin.pl
Mariusz Szczepanik
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: akryczka@golem.umcs.lublin.plbsprus@golem.umcs.lublin.plszczepan@golem.umcs.lublin.pl
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Abstract

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A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T: Ai → Bi, i = 0, 1 is weakly compact, then so is T : Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < P < ∞.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 3 , December 2000 , pp. 389 - 401
Copyright
Copyright © Australian Mathematical Society 2000

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