Commutative Ranges of Analytic Functions in Banach Algebras

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.