Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L(0) that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L(0) has finite codimension in L and there are Lie subalgebras L(0) = L(0) subset of L(1) subset of center dot center dot center dot subset of L(p) = L such that L(i+1) = L(i) + \L(i), L(i+1)\ for all i; (2) L(0) is a Lie ideal of L and dim(L(0)) = infinity. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension. (C) 2008 Elsevier Inc. All rights reserved.