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On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues
- Source :
- Linear and Multilinear Algebra. 68:1310-1328
- Publication Year :
- 2018
- Publisher :
- Informa UK Limited, 2018.
-
Abstract
- We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent.<br />Comment: 18pp Additional results and comments added
- Subjects :
- High Energy Physics - Theory
Primary: 15A16
Pure mathematics
media_common.quotation_subject
logarithms
convergence of the BCH series
FOS: Physical sciences
010103 numerical & computational mathematics
Hwa-Long Gau
Secondary: 34A25
01 natural sciences
Convergnece of the BCH series
High Energy Physics - Phenomenology (hep-ph)
FOS: Mathematics
40A30
Analytic prolongation
0101 mathematics
prolongation of the BCH series
Banach *-algebra
Mathematical Physics
Condensed Matter - Statistical Mechanics
Computer Science::Information Theory
media_common
Mathematics
Algebra and Number Theory
Statistical Mechanics (cond-mat.stat-mech)
Primary: 15A16, 15B99, 40A30. Secondary: 34A25
15B99
analytic prolongation
Baker-Campbell-Hausdorff Theorem
matrix algebras
Prolongation
Mathematics - Rings and Algebras
Mathematical Physics (math-ph)
High Energy Physics - Phenomenology
High Energy Physics - Theory (hep-th)
Rings and Algebras (math.RA)
Baker–Campbell–Hausdorff formula
Identity (philosophy)
Convergence (relationship)
Subjects
Details
- ISSN :
- 15635139 and 03081087
- Volume :
- 68
- Database :
- OpenAIRE
- Journal :
- Linear and Multilinear Algebra
- Accession number :
- edsair.doi.dedup.....b2a4cd27c43f545ab64f04f48ac99a4c