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We give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups in L 1 (R n ) and L 1 (G), for a stratified Lie group G. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0, ). Academic Press

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CYCLIC VECTORS OF INDUCED REPRESENTATIONS a. hulanicki and t. pytlik In this note we prove that for a first countable locally compact group every unitary representation induced by a cyclic representation is cyclic. This result has been recently obtained also by F. Greenleaf and M. Moskowitz [2] in a more complicated way. Let G be a first countable locally compact group. Let Jf be the space of continuous functions with compact support equipped with the Schwartz topology on G. Let D be the cone in Jf' of positive-definite measures on G, i.e., peD if (x**x, p)^0 for all x in Jf. For each p in

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