Level sets and composition operators on the Dirichlet space

Abstract

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.

Compact composition operators on the Dirichlet space and capacity of sets of contact points

“…In view of the result of [6] mentioned in the introduction, if Cap K > 0, there is no hope to find a symbol ϕ such that E ϕ = K and C ϕ is Hilbert-Schmidt on D * . But as was later proved in [5], Cap K > 0 is the only obstruction. We can improve on the results from [5] as follows: our composition operator is not only Hilbert-Schmidt, but in any Schatten class; moreover, we can replace E ϕ = K by E ϕ = E ϕ (1) = K. 2…”

“…But as was later proved in [5], Cap K > 0 is the only obstruction. We can improve on the results from [5] as follows: our composition operator is not only Hilbert-Schmidt, but in any Schatten class; moreover, we can replace E ϕ = K by E ϕ = E ϕ (1) = K. 2…”

“…In particular, it is proved there that if the composition operator C ϕ is Hilbert-Schmidt on D, then the logarithmic capacity Cap E ϕ of E ϕ is 0, but, on the other hand, there are compact composition operators on D for which this capacity is positive. The optimality of this theorem was later proved in [5] under the following form: In this paper, we shall improve on this last result. We prove in Section 4 (Theorem 4.1) that for every compact set K ⊆ ∂D of logarithmic capacity Cap K = 0, there exists a Schur function ϕ ∈ A(D) ∩ D * such that the composition operator C ϕ is in all Schatten classes S p (D * ), p > 0, and for which E ϕ = K (and moreover E ϕ = {e it ; ϕ(e it ) = 1}).…”

Contact points and Schatten composition operators

“…Since C ϕ is bounded on D α , sup ξ∈T µ ϕ,α (W(ξ, h)) = O(h 2+α )(h → 0). see [4,10]. Hence µ ϕ,α (R n, j ) = O(1/2 (2+α)n ).…”

Approximation numbers of composition operators on the Dirichlet space

“…The paper deals with composition operators. This area is widely studied nowadays, on various spaces of analytic functions (Hardy, Bergman, Dirichlet...spaces): one may read for instance the monographs [14] or [4] to get an overview on the subject until the nineties, and [5] or [8] for some recent results in the framework of the Dirichlet space. It seems natural to try to apply again some of the techniques used in the framework of Hardy or Bergman spaces.…”