On commuting operator exponentials
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- by Fotios C. Paliogiannis PDF
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Abstract:
Let $A$, $B$ be bounded operators on a Banach space with $2\pi i$-congruence-free spectra such that $e^Ae^B=e^Be^A$. E. M. E. Wermuth has shown that $AB=BA$. Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for $a,b$ in a complex unital Banach algebra, if the spectrum of $a+b$ is $2\pi i$-congruence-free and $e^ae^b=e^{a+b}=e^be^a$, then $ab=ba$ (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.References
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Additional Information
- Fotios C. Paliogiannis
- Affiliation: Department of Mathematics, St. Francis College, 180 Remsen Street, Brooklyn, New York 11201
- Email: fpaliogiannis@stfranciscollege.edu
- Received by editor(s): April 8, 2002
- Received by editor(s) in revised form: July 2, 2002
- Published electronically: February 24, 2003
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3777-3781
- MSC (2000): Primary 47A60
- DOI: https://doi.org/10.1090/S0002-9939-03-06965-X
- MathSciNet review: 1998185