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On lattice properties of the composition operator

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Abstract

The vector space £b(E) of all order bounded linear operators on a Dedekind complete Riesz space E is both a Riesz space and an algebra. This note investigates the degree of compatibility between the algebraic and lattice structures of £b(E). Two of the main results are the following:

  1. 1.

    An operator on a Banach lattice with an order continuous norm factors through the lattice operations if and only if it is an interval preserving Riesz homotnorphism.

  2. 2.

    A Dedekind complete Banach lattice E has an order continuous norm if and only if 0≤Tn ↑ T in £b(E) implies T 2n ↑ T2.

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References

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Authors and Affiliations

  1. Department of Mathematics, IUPUI, 46205, Indianapolis, IN, USA

    C. D. Aliprantis & O. Burkinshaw

  2. Instytut Matematyczny, Polskiej Akademii Nauk, Poznań, Poland

    P. Kranz

Authors
  1. C. D. Aliprantis
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  2. O. Burkinshaw
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  3. P. Kranz
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Additional information

Research supported in part by NSF grant MCS 81-00787

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Aliprantis, C.D., Burkinshaw, O. & Kranz, P. On lattice properties of the composition operator. Manuscripta Math 36, 19–31 (1981). https://doi.org/10.1007/BF01174810

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  • DOI: https://doi.org/10.1007/BF01174810

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