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

DOI:
https://doi.org/10.1090/S0002-9939-02-06683-2

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 of a region of the complex plane for analytic functions with values in the space of bounded linear operators on a Hilbert space . 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.

**1.**J. Butler,*Perturbation Series for Eigenvalues of Analytic Non-Symmetric Operators*, Arch. Math.**X**(1959), 21-27. MR**21:1535****2.**J. Globevnik and I. Vidav,*A Note on Normal-operator-valued Analytic Functions*, Proc. Amer. Math. Soc.**37-2**(1973), 619- 621.MR**46:9761****3.**S. L. Jamison,*Perturbation of normal operators*, Proc. Amer. Math. Soc.**5**(1954), 103-110. MR**15:720c****4.**M. Reed and B. Simon,*Methods of Modern Mathematical Physics. IV Analysis of Operators*, Methods of Modern Mathematical Physics, Academic Press, New York, 1978. MR**58:12429c****5.**W. Rudin,*Real and Complex Analysis*, Second Edition. McGraw-Hill, 1974.MR**49:8783****6.**F. Wolf,*Analytic perturbation of operators in Banach spaces*, Math. Ann.**124**(1952), 317-333. MR**14:288c**

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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:
https://doi.org/10.1090/S0002-9939-02-06683-2

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