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Uniqueness of Taylor's functional calculus

Author: Mihai Putinar
Journal: Proc. Amer. Math. Soc. 89 (1983), 647-650
MSC: Primary 47A60; Secondary 32C35, 46H30
MathSciNet review: 718990
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Abstract: Two uniqueness results concerning Fréchet module structures over algebras of holomorphic functions defined on some complex manifolds are presented, containing as particular cases uniqueness theorems for J. L. Taylor's analytic functional calculi for commuting $ n$-tuples of linear continuous operators on Fréchet spaces [7], [9]. Namely, the first statement says that the Spectral Mapping Theorem insures the unicity of the functional calculus and thus it improves Zame's unicity theorem [11, Theorem 1], while the second statement gives a unicity condition which is an analogue of the compatibility property [3, Theorem I.4.1] in spectral theory of several variables in commutative Banach algebras. As a corollary the two functional calculi constructed in [7] and [9] by J. L. Taylor coincide.

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Keywords: Analytic functional calculus, linear continuous operators, J. L. Taylor's joint spectrum, Fréchet module structure, Riemann domain, Stein space, commuting $ n$-tuple
Article copyright: © Copyright 1983 American Mathematical Society

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