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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

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


Author: Daniel Turcotte
Journal: Proc. Amer. Math. Soc. 131 (2003), 1399-1404
MSC (2000): Primary 46K05
Published electronically: December 16, 2002
MathSciNet review: 1949869
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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.


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Additional Information

Daniel Turcotte
Affiliation: Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, Québec, Canada H3C 3J7
Address at time of publication: 5946 Dalebrook Crescent, Mississauga, Ontario, Canada L5M 5S1
Email: daniel_turcotte@sympatico.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06683-2
PII: S 0002-9939(02)06683-2
Keywords: Propagation of normality, Banach algebra with continuous involution, propagation property within Banach subspaces
Received by editor(s): June 1, 2000
Received by editor(s) in revised form: October 9, 2001
Published electronically: December 16, 2002
Additional Notes: The results contained in this paper are part of the author’s Ph.D. thesis written while a guest at the Université of Montréal. Translation from French into English of the present work and improvements in the proof of the Propagation of Normality Theorem were done during his Postdoctorate at The University of Toronto. The first draft of this paper was written at Ryerson Polytechnic University. The author thanks the referee for the detailed stylistic comments that were provided.
Communicated by: David R. Larson
Article copyright: © Copyright 2002 Daniel Turcotte, 5946 Dalebrook Crescent, Mississauga, Ontario, Canada L5M 5S1