Analysis Seminar

Speaker: Stefan Richter

Title: Cyclicity and iterated Logarithms in the Dirichlet space

Abstract: If $\mathcal H$ denotes a Hilbert space of analytic functions in the open unit disc $\mathbb D\subseteq \mathbb C$, then a function $f\in \mathcal H$ is called cyclic, if the polynomial multiples of $f$ are dense in $\mathcal H$. Cyclic functions cannot have any zeros in the open unit disc, and for many spaces $\mathcal H$ it is interesting to characterize cyclicity in terms of how small $|f(z)|$ can be near the boundary of $\mathbb D$. For example, if $\mathcal H=H^2$, the space of analytic functions in $\mathbb D$ whose Taylor coefficients are square summable, then $f$ is cyclic in $H^2$, if and only if it is a so-called outer function, i.e. the harmonic function $\log|f(z)|$ equals the Poisson integral of an absolutely continuous measure. This excludes for example functions of the type $f(z)=e^{-\frac{1+z}{1-z}}$ where $|f|$ has exponential decline on the radius $\to 1$, and $\log |f(z)|$ equals the Poisson integral of a point mass.

Let $D={f\in \mathrm{Hol}(\mathbb D): \int{\mathbb D} |f'|^2dxdy<\infty} $ denote the Dirichlet space of the unit disc $\mathbb D$. One checks that $D\subseteq H^2$ with $|f|{H^2}\le |f|D$. It follows that cyclic functions in $D$ must be outer, but deciding which outer functions are cyclic in $D$ is an open problem. We show that an outer function $f\in D$ is cyclic in $D$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D)$. If $f$ is a bounded analytic function in $D$, then cyclicity can also be checked via iterated logarithms. For example, we show that an outer function $f$ is cyclic, whenever $\log(1+ \log(1/f))\in N^+(D)$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity. The talk is based on joint work with Alexandru Aleman.