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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 ...
The aim is to investigate certain spectral properties, such as decomposability, the spectral mapping property and the Lyubich-Matsaev property, for linear differential operators with constant coefficients ( and more general Fourier multiplier operators) acting in . The criteria developed for such operators are quite general and p-dependent, i.e. they hold for a range of p in an interval about 2 (which is typically not (1,∞)). The main idea is to construct appropriate functional calculi: this is...
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