| dc.description.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 |
en_US |