Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees

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*} \kappa_{\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.