On the Brown--Shields conjecture for cyclicity in the Dirichlet space

Cyclicity in Dirichlet-type spaces and extremal polynomials

For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\|p f-1\|_α$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\|p_{n}f-1\|_α$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.

Cyclicity and invariant subspaces in the Dirichlet spaces

Let $μ$ be a positive finite measure on the unit circle and $\mathcal{D} (μ)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (μ)$ is cyclic if and only if $c\_μ(Z (f))= 0$, where $c\_μ$ is the capacity associated with $\mathcal{D} (μ)$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.

Hardy算子：追随Brown、Halmos和Shields的脚步

Hardy operators: In the footsteps of Brown, Halmos, and Shields

本文研究两个经典Hardy算子，构造显式循环向量并刻画不变子空间，完整描述框架向量。

作为de Branges-Rovnyak空间的无限支撑调和加权Dirichlet空间

Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces

本文研究调和加权Dirichlet空间与de Branges-Rovnyak空间的相等问题，突破有限支撑限制，刻画了无限支撑测度下两类空间相等的条件，还建立了相关T(1)型结果与扰动定理，并证明了Brown-Shields猜想。