Commutators on \((\sum\ell_q)_{\ell_1}\)
Let \(X, Y, Z\) be Banach spaces. A bounded linear operator \(T: X \to Y\) is said to be \(Z\)-strictly singular if there is no subspace \(Z_0\) of \(X\) which is isomorphic to \(Z\) and such that the restriction map \(T \mid_{Z_0}\) is an isomorphism. In the present paper, the author considers the case where \(Y=X= (\sum l_q)_{l_1}\) with \(1 \leq q < \infty\). Firstly, he shows that the unique maximal ideal in \(X\) is the set of all \(X\)-strictly singular operators. Secondly, he shows that \(T\) is a commutator if and only if for every nonzero \(\lambda \in \mathbb C\), \(T-\lambda I\) is not \(X\)-strictly singular. It should be pointed out that this paper is a continuation and extension of earlier work (see [\textit{D.-Y. Chen} et al., Stud. Math. 206, No. 2, 175--190 (2011; Zbl 1232.47031); corrigendum ibid. 223, No. 2, 187--191 (2014; Zbl 1302.47054)]).
scientific article; zbMATH DE number 6420185