Infinite-dimensional triangularizable algebras

Abstract

Abstract Let End k ⁢ ( V ) {\mathrm{End}_{k}(V)} denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define X ⊆ End k ⁢ ( V ) {X\subseteq\mathrm{End}_{k}(V)} to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of End k ⁢ ( V ) {\mathrm{End}_{k}(V)} is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of End k ⁢ ( V ) {\mathrm{End}_{k}(V)} , which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.