Roots and logarithms of multipliers

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By now it is a well-known fact that if f f is a multiplier for the Drury-Arveson space H n 2 H^2_n , and if there is a c > 0 c > 0 such that | f ( z ) | ≥ c |f(z)| \geq c for every z ∈ B z \in \mathbf {B} , then the reciprocal function 1 / f 1/f is also a multiplier for H n 2 H^2_n . We show that for such an f f and for every t ∈ R t \in \mathbf {R} , f t f^t is also a multiplier for H n 2 H^2_n . We do so by deriving a differentiation formula for R m ( f t h ) R^m(f^th) . Moreover, by this formula the same result holds for spaces H m , s \mathcal {H}_{m,s} of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier f f of H n 2 H^2_n , log ⁡ f \log f is a multiplier of H n 2 H^2_n if and only if log ⁡ f \log f is bounded on B \mathbf {B} .