Uniqueness of Taylor’s functional calculus

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 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.