Commutators on \(\ell _1\)
The classification of commutators, that is, of elements of the form \(AB-BA\) in a given Banach algebra, has a long history. Restricting ourselves to the case of \(B(X)\), the algebra of all operators on the Banach space \(X\), the first breakthrough was made by \textit{A.\,Brown} and \textit{C.\,Pearcy} [Ann.\ Math.\ (2) 82, 112--127 (1965; Zbl 0131.12302)] who proved that for a Hilbert space \(H\), the only operators on \(H\) that are not commutators are those of the form \(\lambda I + K\), with \(K\) compact. \textit{C.\,Apostol} [Rev.\ Roum.\ Math.\ Pures Appl.\ 17, 1513--1534 (1972; Zbl 0247.47030)] extended the Brown--Pearcy result to \(X = \ell_p\) with \(1 < p < \infty\). The main result of the present paper is that the commutators on \(\ell_1\) are the operators not of the form \(\lambda I + K\), with \(\lambda \neq 0\) and \(K\) compact. Other results are also proved.
scientific article; zbMATH DE number 5565765
scientific article