Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff
type: Item
instance of: scholarly article
title: Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff
zbMATH Open document ID: 0934.46053
author: Wojciech Wojtyński
published in: Journal of Functional Analysis
publication date: 2000-04-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 1215691
zbMATH Keywords: Banach-Lie algebras
zbMATH Keywords: quasinilpotence
zbMATH Keywords: Baker-Campbell-Hausdorff series
zbMATH Keywords: Engel theorem
zbMATH Keywords: quasinilpotent operators
reviewed by: A. Castellón Serrano
MaRDI profile type: Publication
full work available at URL: jfan.1997.3202
OpenAlex ID: W2011472148
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DOI: 10.1006/JFAN.1997.3202
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description: scientific article; zbMATH DE number 1215691