Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals
The article is devoted to Lie and Banach Lie algebras \({\mathcal L}\) of bounded linear operators on an infinite-dimensional complex Banach space \(X\). Their subclass with Lie subalgebras \({\mathcal L}_0\) having a nontrivial closed invariant subspace in \(X\) of finite codimension or dimension is studied. A Lie subalgebra \({\mathcal L}_0\) of a Lie algebra \(\mathcal L\) is called nondegenerate in \(\mathcal L\), if \({\mathcal L}_0\) has finite codimension in \(\mathcal L\) and \({\mathcal L} = {\mathcal L}_0 + [{\mathcal L}_0, {\mathcal L}]\). If a chain of embedded Lie subalgebras \({\mathcal L}_0 = {\mathcal L}^0 \subseteq {\mathcal L}^1 \subseteq \dots \subseteq {\mathcal L}^p = {\mathcal L}\) is such that \({\mathcal L}^j\) is nondegenerate in \({\mathcal L}^{j+1}\) for each \(j\), then \({\mathcal L}_0\) is called related to \(\mathcal L\). One the main results of the paper is the following Theorem 8.1. Let \({\mathcal L}_0 \subseteq L \subseteq {\mathcal L}\) be Lie subalgebras of an algebra \({\mathcal B}(X)\) of all bounded operators on an infinite-dimensional complex Banach space X and let \(\dim ({\mathcal L}_0) = \infty \). Suppose that \({\mathcal L}_0\) has a closed invariant subspace \(X_0\). {\parindent=6mm \begin{itemize}\item[(i)] Let \({\mathcal L}_0\) be related to \(L\) and let \(L\) be a Lie ideal of \({\mathcal L}\). If \(\text{codim} (X_0)<\infty \) or \(\dim(X_0)<\infty \), then \(\mathcal L\) is reducible. \item[(ii)] Let \({\mathcal L}_0\) be a Lie ideal of \(L\), let also \(\text{codim}({\mathcal L}_0)<\infty \) and let \(L\) be related to \(\mathcal L\). If \(\text{codim}(X_0)<\infty \) and \({\mathcal L}_0\) has no operator-nilpotent Lie ideals of finite codimension, then \(\mathcal L\) is reducible. \end{itemize}} Moreover, invariant subspaces of operator Lie algebras are investigated. These results are applied to studies of Lie ideals in Banach Lie algebras, particularly, to the problem whether there exist nontrivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra \({\mathcal L}\) which contains a nontrivial closed Lie subalgebra of finite codimension.
scientific article; zbMATH DE number 5522892
scientific article