局部有限根有向树的Dirichlet空间
Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees
摘要：
设$\mathscr T=(V, \mathcal E)$是一棵无叶的、局部有限根的有向树。我们将$\mathscr T$与一个单参数Dirichlet空间族$\mathscr H_q~(q \geqslant 1)$联系起来，它是定义在复平面单位圆盘$\mathbb D$上的向量值全纯函数的Hilbert空间。这些空间可以被实现为与正定核\begin{eqnarray*} κ_{\mathscr H_q}(z, w) = \sum_{n=0}^{\infty}\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf{root}}\rangle} + \sum_{v \in V_{\prec}} \sum_{n=0}^{\infty} \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{eqnarray*}相关联的再生核希尔伯特空间，其中$V_{\prec}$表示$\mathscr T$的分支顶点的集合，$n_v$表示$v \in V$在$\mathscr T,$中的深度，并且$P_{\langle e_{\mathsf{root}}\rangle}$、$~P_{v}~(v \in V_{\prec})$是某些正交投影。进一步讨论了$\mathscr H_q.$上的$z$乘算子$\mathscr M_{z, q}$的结构性质。进一步讨论了$\mathscr H_q.$上的$z$乘算子$\mathscr M^{(1)}_z$和$\mathscr M^{(2)}_z$在分别对应于有向树$\mathscr T_1$和$\mathscr T_2$的Dirichlet空间$\mathscr H_q$上的酉等价问题。
英文摘要：
Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $\mathbb D$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel \begin{eqnarray*} κ_{\mathscr H_q}(z, w) = \sum_{n=0}^{\infty}\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf{root}}\rangle} + \sum_{v \in V_{\prec}} \sum_{n=0}^{\infty} \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{eqnarray*} where $V_{\prec}$ denotes the set of branching vertices of $\mathscr T$, $n_v$ denotes the depth of $v \in V$ in $\mathscr T,$ and $P_{\langle e_{\mathsf{root}}\rangle}$, $~P_{v}~(v \in V_{\prec})$ are certain orthogonal projections. We also discuss some structural properties of the operator $\mathscr M_{z, q}$ of multiplication by $z$ on $\mathscr H_q.$ Further, we discuss the question of unitary equivalence of operators $\mathscr M^{(1)}_z$ and $\mathscr M^{(2)}_z$ of multiplication by $z$ on Dirichlet spaces $\mathscr H_q$ associated with directed trees $\mathscr T_1$ and $\mathscr T_2$ respectively.
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