LAUSR

Let $$\mathscr {T}=(V, \mathcal E)$$T=(V,E) be a leafless, locally finite rooted directed tree. We associate with $$\mathscr {T}$$T a one parameter family of Dirichlet spaces $$\mathscr {H}_q~(q \geqslant 1)$$Hq(q⩾1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $$\mathbb D$$D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel $$\begin{aligned} \kappa _{\mathscr {H}_q}(z, w)= & {} \sum _{n=0}^{\infty }\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf {root}}\rangle } \\&+\,\sum _{v \in V_{\prec }} \sum _{n=0}^{\infty } \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{aligned}$$κHq(z,w)=∑n=0∞(1)n(q)nznw¯nP⟨eroot⟩+∑v∈V≺∑n=0∞(nv+2)n(nv+q+1)nznw¯nPv(z,w∈D),where $$V_{\prec }$$V≺ denotes the set of branching vertices of $$\mathscr {T}$$T, $$n_v$$nv denotes the depth of $$v \in V$$v∈V in $$\mathscr {T},$$T, and $$P_{\langle e_{\mathsf {root}}\rangle }$$P⟨eroot⟩, $$~P_{v}~(v \in V_{\prec })$$Pv(v∈V≺) are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators $$\mathscr {M}^{(1)}_z$$Mz(1) and $$\mathscr {M}^{(2)}_z$$Mz(2) of multiplication by z on Dirichlet spaces $$\mathscr {H}_q$$Hq associated with directed trees $$\mathscr {T}_1$$T1 and $$\mathscr {T}_2$$T2 respectively.