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Let be a tuple of commuting operators on a Banach space . We
discuss various conditions equivalent to that the holomorphic (Taylor) functional
calculus has an extension to the real-analytic functions or various ultradifferentiable
classes. In particular, we discuss the possible existence of a functional calculus for
smooth functions. We relate the existence of a possible extension to existence of a
certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of
. If ...
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