Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees

Let $\mathcal{T}=(V,\mathcal{E})$ be a leafless, locally finite rooted directed tree. We associate with $\mathcal{T}$ a one parameter family of Dirichlet spaces ${\mathcal{H}}_q(q⩾1)$, 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{$\mathsf{r}\mathsf{o}\mathsf{o}\mathsf{t}$}\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 $\mathcal{T}$, $n_v$ denotes the depth of $v\in V$ in $\mathcal{T},$ and $P_{⟨e_{\mathsf{r}\mathsf{o}\mathsf{o}\mathsf{t}}⟩}$, $P_v(v\in V_{\prec })$ are certain orthogonal projections. We also discuss some structural properties of the operator ${\mathcal{M}}_{z,q}$ of multiplication by $z$ on ${\mathcal{H}}_q\mathrm{.}$ Further, we discuss the question of unitary equivalence of operators ${\mathcal{M}}_z^{(1)}$ and ${\mathcal{M}}_z^{(2)}$ of multiplication by $z$ on Dirichlet spaces ${\mathcal{H}}_q$ associated with directed trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ respectively.