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On commuting operator exponentials

Author: Fotios C. Paliogiannis
Journal: Proc. Amer. Math. Soc. 131 (2003), 3777-3781
MSC (2000): Primary 47A60
Published electronically: February 24, 2003
MathSciNet review: 1998185
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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.

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Additional Information

Fotios C. Paliogiannis
Affiliation: Department of Mathematics, St. Francis College, 180 Remsen Street, Brooklyn, New York 11201

Keywords: Commuting exponentials, holomorphic functional calculus, Borel functional calculus
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
Article copyright: © Copyright 2003 American Mathematical Society