Roots and Logarithms of Multipliers

TL;DR

This paper explores the properties of multipliers in the Drury-Arveson space, demonstrating that powers and logarithms of such multipliers are also multipliers under certain conditions, using a differentiation formula.

Contribution

It introduces a differentiation formula for powers of multipliers, extending known properties to a broader class of functions and spaces.

Findings

f^t is a multiplier for all real t if f is a bounded multiplier

log f is a multiplier iff log f is bounded on B

The results extend to Besov-Dirichlet type spaces

Abstract

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 $B$.