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Positive operators as commutators of positive operators

Volume 245 / 2019

Roman Drnovšek, Marko Kandić Studia Mathematica 245 (2019), 185-200 MSC: Primary 47B07, 47B47; Secondary 47B65, 46B42. DOI: 10.4064/sm170703-26-9 Published online: 20 July 2018

Abstract

It is known that a positive commutator $C=A B - B A$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. We study the question under which conditions a positive operator can be written as a commutator of positive operators. As a special case of our main result we find that positive compact operators on order continuous Banach lattices which admit order Pełczyński decomposition are commutators of positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L^p(\mu )$ $(1 \lt p \lt \infty )$.

Authors

  • Roman DrnovšekFaculty of Mathematics and Physics
    University of Ljubljana
    Jadranska 19
    1000 Ljubljana, Slovenia
    e-mail
  • Marko KandićFaculty of Mathematics and Physics
    University of Ljubljana
    Jadranska 19
    1000 Ljubljana, Slovenia
    e-mail

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