AI 中文总结
本文研究调和加权Dirichlet空间，刻画了内乘子、算子相似性及不变子空间，推广了Richter的结果，并利用Richter-Sundberg公式解决关键问题。
AI 中文摘要
我们考虑由Borel测度诱导的调和加权Dirichlet空间$D(\mu)$，这些测度与单位圆$\mathbb T$上的Lebesgue测度$m$相互绝对连续。设$H^2$和$D$分别表示单位圆盘$\mathbb D$上的Hardy空间和Dirichlet空间。对于任意$f\in H^2$，令$m_f$表示由$dm_f(\zeta)=|f(\zeta)|^2 dm(\zeta)$定义的测度。我们的第一个结果刻画了当$f$满足条件$|f'(z)|^2=O(\frac{1}{(1-|z|)^{1-\epsilon}})$（对某个$\epsilon>0$）时，$D(m_f)$中函数的特征。然后我们研究这些空间的乘子代数，特别关注内乘子。具体地，当$f'$有界时，我们利用$f$的径向零点集给出了$D(m_f)$的乘子代数中所有奇异内函数的显式描述。作为应用，我们证明如果$f',g'$有界且$f$和$g$的径向零点集不同，则算子$(M_z, D(m_f))$和$(M_z, D(m_g))$不相似。这推广了Richter的一个结果，该结果表明算子$(M_z, D(m_{z-1}))$和$(M_z, D)$不相似。然后，我们分类了$(M_z, D(\mu))$的所有不变子空间$\mathcal M$，使得$M_z|_{\mathcal M}$与$(M_z, D(\mu))$相似，其中$\mu$是与$m$相互绝对连续的任意测度。最后，我们研究了一个特殊问题：找出所有$g\in D$使得$gD(m_g)$是$(M_z, D)$的闭不变子空间。我们许多结果的一个关键成分是局部Dirichlet积分的Richter-Sundberg公式。
英文摘要
We consider the harmonically weighted Dirichlet spaces $D(μ)$ induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure $m$ on the unit circle $\mathbb T$. Let $H^2$ and $D$ denote the Hardy space and the Dirichlet space on the unit disc $\mathbb D$, respectively. For any $f\in H^2$, let $m_f$ denote the measure defined by $dm_f(ζ)=|f(ζ)|^2 dm(ζ)$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $|f'(z)|^2=O(\frac{1}{(1-|z|)^{1-ε}})$ for some $ε>0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f'$ is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of $D(m_f)$ in terms of the radial zero set of $f$. As an application, we prove that if $f',g'$ are bounded and the radial zero sets of $f$ and $g$ are different, then the operators $(M_z, D(m_f))$ and $(M_z, D(m_g))$ are not similar. This generalizes a result of Richter showing that the operators $(M_z, D(m_{z-1}))$ and $(M_z, D)$ are not similar. Then, we classify all invariant subspaces $\mathcal M$ of $(M_z, D(μ))$ for which $M_z|_{\mathcal M}$ is similar to $(M_z, D(μ))$, where $μ$ is any measure mutually absolutely continuous with respect to $m$. Finally, we study a special case of the problem of finding all functions $g\in D$ for which $gD(m_g)$ is a closed invariant subspace of $(M_z, D)$. A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.
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