Weighted Composition Operators on Hardy Spaces

By Manuel D. Contreras, Alfredo García Hernández-Díaz. 2001

102 citations recorded by the source corpus.

Let ϕ, ψ be analytic functions defined on D, such that ϕ(D) ⊆ D. The operator given by f ↦ ψ(f ∘ ϕ) is called a weighted composition operator. In this paper we deal with the boundedness, compactness, weak compactness, and complete continuity of weighted composition operators on Hardy spaces Hp (1 ≤ p < ∞). In particular, we prove that such an operator is compact on H1 if and only if it is weakly compact on this space. This result depends on a technique which passes the weak compactness from an operator T to operators dominated in norm by T.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34ef583fcd95949b0d727e664e21176fb050bce6

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In Applied Mathematics

1. Z. C̆uc̆ković
2. W. Smith
3. J. H. Shapiro
4. R. Zhao
5. Trieu Le
6. S. Stević
7. Mikael Lindström
8. Pekka J. Nieminen

In Mathematical Physics

1. Z. C̆uc̆ković
2. W. Smith
3. J. H. Shapiro
4. R. Zhao

Citations in the corpus

• Hardy spaces that support no compact composition operators (2003, methodology): “4 of [2]: For certain weights its hypotheses are satisfied for no holomorphic selfmap φ of U.”
• Weighted Composition Operators on the Bergman Space (2004, methodology): “The boundedness and compactness of weighted composition operators on the Hardy spaces were characterized in [3], using a Carleson measure condition.”