Invariant subspaces of operator Lie algebras and ideals of Banach Lie algebras

We consider some problems on the existence and structure of invariant subspaces for operator Lie algebras that possess ideals or massive subalgebras with special properties (e.g. Volterra ideals). We will discuss also some applications to the structure theory of Banach-Lie algebras – in particular the existence of closed ideals of finite codimension in Banach-Lie algebras that have closed Lie subalgebras of finite codimension. Main results of the report were obtained in joint works with E.Kissin and Yu.V. Turovskii.