ROOTS AND LOGARITHMS OF MULTIPLIERS

By now it is a well-known fact that if f is a multiplier for the Drury-Arveson space H_n² , and if there is a c > 0 such that |f(z)| ≥ c for every z ∈ B, then the reciprocal function 1/f is also a multiplier for H_n² . We show that for such an f and for every t ∈ R, f^t is also a multiplier for H_n² . We do so by deriving a differentiation formula for R^m(f^th). Moreover, by this formula the same result holds for spaces Hm,s of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier f of H_n² , log f is a multiplier of H_n² if and only if log f is bounded on B.