Weighted Composition Operators on the Space of Bounded Harmonic Functions

Abstract

We determine the norm and the essential norm of the difference of weighted composition operators on the space of bounded harmonic functions on the open unit disk. The argument is done on the boundary. Mathematics Subject Classification (2010). Primary 47B38; Secondary 30H10.Keywords. Weighted composition operator, the space of bounded harmonic functions, essential norm.

Boundary vs. interior conditions associated with weighted composition operators

“…We give direct proofs for the corresponding equivalences in Section 4. In Section 5, we give another proof of the estimate for the essential norm C − C ψ ∞ e obtained in [2] applying the results given in [10]. That is, we prove the main result (Theorem 5.2) using the behavior of and ψ on the open unit disk instead of on the maximal ideal space of H ∞ .…”

“…In [10], the authors characterized compactness of M C and differences M C −M C ψ , and provided formulas for operator and essential norms of differences M C − M C ψ in terms of certain conditions on functions ψ on the boundary ∂D.…”

“…Let ∈ H ∞ and ψ ∈ S(D). For compactness of M C − M C ψ on H ∞ , the interior condition on functions ψ is given in [9] and the boundary condition on them is given in [10]. So these interior and boundary conditions are equivalent.…”

“…So these interior and boundary conditions are equivalent. In the next section we recall properties of L ∞ (∂D) functions on ∂D investigated in [10] and give a direct proof of the mentioned equivalence in Section 3. Similar situations occur for other properties of weighted composition operators on ∞ and H ∞ .…”

“…In order to study L ∞ (∂D) functions, we recall some elementary facts and notations used in [10]. For ∈ ∞ , let R be the set of θ ∈ ∂D at which has a radial limit.…”