Propagation of normality along regular analytic Jordan arcs in analytic functions with values in a complex unital Banach algebra with continuous involution

Abstract:

Globevnik and Vidav have studied the propagation of normality from an open subset $V$ of a region $\mathcal {D}$ of the complex plane for analytic functions with values in the space $\mathcal {L}(\mathcal {H})$ of bounded linear operators on a Hilbert space $\mathcal {H}$. We obtain a propagation of normality in the more general setting of a converging sequence located on a regular analytic Jordan arc in the complex plane for analytic functions with values in a complex unital Banach algebra with continuous involution. We show that in this more general setting, the propagation of normality does not imply functional commutativity anymore as it does in the case studied by Globevnik and Vidav. An immediate consequence of the Propagation of Normality Theorem is that the further generalization given by Wolf of Jamison’s generalization of Rellich’s theorem is equivalent to Jamison’s result. We obtain a propagation property within Banach subspaces for analytic Banach space-valued functions. The propagation of normality differs from the propagation within Banach subspaces since the set of all normal elements does not form a Banach subspace.