Holomorphic Hörmander-type functional calculus on sectors and strips

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.