Commutative ranges of analytic functions in Banach algebras
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- by R. J. Fleming and J. E. Jamison PDF
- Proc. Amer. Math. Soc. 93 (1985), 48-50 Request permission
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.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- 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