Invariant subspaces of operator Lie algebras and the theory of \(K\)-algebras (Q1812462)

Let \(B(X)\) denote the space of bounded linear operators on a Banach space \(X\). A closed subspace \( Y \subset X\) is said to be invariant for a set \(M \subset B(X)\) if \(T(Y) \subset Y\) for all \(T \in M\). It is said to be hyperinvariant if it is invariant for \(M\) and its commutator. Considering \(B(X)\) as a Lie algebra under \([T,S] = TS-ST\), two of the results announced in this paper are: 1. If a closed Lie algebra of compact operators contains a nonzero ideal having no nonzero finite rank operator, then this algebra has a nontrivial invariant subspace. 2. If a Lie algebra of compact operators contains a nonzero ideal of quasinilpotent operators (also called Volterra ideal), then it has a nontrivial hyperinvariant subspace.