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# Commutative Ranges of Analytic Functions in Banach Algebras

R. J. Fleming and J. E. Jamison
Proceedings of the American Mathematical Society
Vol. 93, No. 1 (Jan., 1985), pp. 48-50
DOI: 10.2307/2044551
Stable URL: http://www.jstor.org/stable/2044551
Page Count: 3
Let A denote a complex unital Banach algebra with Hermitian elements H(A). We show that if F is an analytic function from a connected open set D into A such that F(z) is normal $(F(z) = u(z) + iv(z),\quad\text{where}\quad u(z), v(z) \in H(A)\quad\text{and}\quad u(z)v(z) = v(z)u(z))$ for each z ∈ D, then F(z)F(w) = F(w)F(z) for all w, z ∈ 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.