Similarity invariant subspaces and Lie ideals in operator algebras. (Q1865918)

Let \(X\) be a complex Banach space and \(L(X)\) be the algebra of all linear bounded operators \(T: X \rightarrow X\) endowed with the uniform operator topology. \(GL(X)\) denotes the group of all invertible operators belonging to \(L(X)\). A closed subspace \(J\) of \(L(X)\) is called a Lie ideal if \([A,B]=AB -BA \in J\) for all \(A \in J\) and \(B \in L(X)\). In this paper, the authors obtain a full description of nontrivial Lie ideals (and hence a complete list of \(GL(X)\)-invariant subspaces) for the spaces \(l^{p}\) \((1 \geq p < {\infty})\) of \(p\)-summable sequences and the space \(c_{0}\) of sequences converging to zero.