Cyclicity and iterated logarithms in the Dirichlet space

Abstract

Let $D(\hat{I}¼)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb{D}$. We show that outer functions $f\in D(\hat{I}¼)$ are cyclic in $D(\hat{I}¼)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^{+}(D(\hat{I}¼))$. If $f$ has $H^{\infty }$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log (1+\log (1/f))\in N^{+}(D(\hat{I}¼))$. This condition can be checked by verifying that $\log (1+\log (1/f))\in D(\hat{I}¼)$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity.

9 pages, 1 figure