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Representation of Banach Ideal Spaces and Factorization of Operators

Published online by Cambridge University Press:  20 November 2018

Evgenii I. Berezhnoĭ
Affiliation:
Department of Mathematics, Yaroslavl’ State University, Sovetskaya 14, 150 000 Yaroslavl’, Russia, email: ber@uniyar.ac.ru
Lech Maligranda
Affiliation:
Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden, email: lech@sm.luth.se website: www.sm.luth.se/∽lech/
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Abstract

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Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2005

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