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Quantitative approach to weak noncompactness in the polygon interpolation method

Published online by Cambridge University Press:  17 April 2009

Andrzej Kryczka
Andrzej Kryczka
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland, e-mail: andrzej.kryczka@umcs.lublin.pl
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We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach N-tuples. In the case of operators acting between two J-spaces or two K-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 49 - 62
Copyright
Copyright © Australian Mathematical Society 2004

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