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

摘要

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 Li+1 = L-i + L-i, Li+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.