Composition operators on a local Dirichlet space

Semigroups of Composition Operators on Local dirichlet Spaces

“…By [5,Theorem 1], α ∈ R and, for every t ≥ 0, the angular derivative ϕ t (ζ) exists with ϕ t (ζ) = e αt . Thus, C t is bounded for every t > 0 [12,Theorem 2]. Therefore,…”

“…By the subordination principle, composition operators act continuously on the Hardy spaces H p for 1 ≤ p < ∞ [7]. Recently, Sarason and Silva [12] studied composition operators on the Dirichlet space D(µ). They used counting functions along with Bergman embedding theorems to describe when C ϕ is bounded or compact on D(µ).…”

“…In this section we consider composition operator semigroups {C t } acting on D(δ ζ ). Following the work of Sarason and Silva [12] on composition operators, we will distinguish two cases. The first case is when ζ is a fixed point for some member (and hence for every member [4]) of the family {ϕ t : t ∈ (0, ∞)} and the second case is when |ϕ t (ζ)| < 1 for every t > 0.…”

Composition Operators on a Class of Analytic Function Spaces Related to Brennan’s Conjecture

“…The spaces C + (λ − z)H 2 (D), λ ∈ ∂D are called local Dirichlet spaces and were introduced in [7]. Composition operators on local Dirichlet spaces are studied in [9]. The spaces C + (λ − z) It is easy to establish the following connection between composition operators on the spaces studied in this section and those acting on the spaces S α .…”

Weighted Composition Operators on H 2 and Applications

“…Thus ϕ is an element of D(δ 1 ) having horocyclic limit equal to 1 at 1. Furthermore, ϕ induces a bounded composition operator on D(δ 1 ), [21,Theorem 2].…”