Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff
The aim of this paper is to prove the equivalence, in a real Banach-Lie algebra \(L\), between the quasinilpotence of all its adjoint operators \(ad_a\) and the convergence of the Baker-Campbell-Hausdorff series for any \(a,b\in L\). We recall that the Baker-Campbell-Hausdorff series is the real power series of nonconmutative formal variables \(x\) and \(y\) obtained by the composition \(\Theta =W\circ Z\), where \(W\) and \(Z\) are given by \[ W(z)=\log (1+z)=\sum_{n=1}^\infty (-1)^{n+1}{z^n\over n},\quad Z(x,y)=e^xe^y-1=\sum_{j+k\geq 1}{s^j\over j!}{y^k\over k!}. \] Several open questions are proposed, in particular a generalization of the Engel theorem: If \(L\) is a quasi-nilpotent Banach-Lie subalgebra of the Banach algebra \(B(X)\) of all continuous endomorphisms of a Banach space \(X\), then the associative subalgebra of \(B(X)\) generated by \(L\) is also composed of quasinilpotent operators.
scientific article; zbMATH DE number 1215691