Roots and Logarithms of Multipliers

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