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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Commutative ranges of analytic functions in Banach algebras


Authors: R. J. Fleming and J. E. Jamison
Journal: Proc. Amer. Math. Soc. 93 (1985), 48-50
MSC: Primary 46H99; Secondary 46H30, 47A05
DOI: https://doi.org/10.1090/S0002-9939-1985-0766525-6
MathSciNet review: 766525
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Abstract: Let $ A$ denote a complex unital Banach algebra with Hermitian elements $ (A)$. We show that if $ F$ is an analytic function from a connected open set $ D$ into $ A$ such that $ F(z)$ is normal $ (F(z) = u(z) + i\upsilon (z)$, where $ u(z)$, $ \upsilon (z) \in H(A)$ and $ u(z)\upsilon (z) = \upsilon (z)u(z))$ for each $ z \in D$, then $ F(z)F(w) = F(w)F(z)$ for all $ w$, $ z \in D$. This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0766525-6
Keywords: Unital Banach algebras, Hermitian elements, normal elements, essentially normal
Article copyright: © Copyright 1985 American Mathematical Society