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Commutators on \(L_p\), \(1\leq p <\infty\). (English) Zbl 1278.47038

The problem of describing operators in \({\mathcal L}(X)\) that are commutors has a long and rich history. The paper under review gives a definitive answer for the case where \(X=L_p = L_p([0,1],\mu)\), \(1\leq p <\infty\), \(\mu\) being the Lebesgue measure. It states that an operator \(T\in {\mathcal L}(L_p)\) is a commutator if and only if, for every \(\lambda \neq 0\), the operator \(T-\lambda I\) does not belong to the largest ideal of \({\mathcal L}(L_p)\).

MSC:

47B47 Commutators, derivations, elementary operators, etc.
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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