Dirichlet Spaces Associated with Locally Finite Rooted Directed Trees

Abstract

Let T = (V, E) be a leafless, locally finite rooted directed tree. We associate with T a one parameter family of Dirichlet spaces Hq (q 1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel κH q (z,w) = ∞

n=0 (1)n (q)n znwn P eroot + v∈V≺ ∞

n=0 (nv + 2)n (nv + q + 1)n znwn Pv (z,w ∈ D), where V≺ denotes the set of branching vertices of T , nv denotes the depth of v ∈ V in T , and P eroot , Pv (v ∈ V≺) are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators M(1) z and M(2) z of multiplication by z on Dirichlet spaces Hq associated with directed trees T1 and T2 respectively