Zeros Under Unitary Weighted Composition Operators in the Hardy and Bergman Spaces

Abstract

AbstractLet $$\mathbb {D}$$ D denote the unit disk of the complex plane and let $$\mathcal {A}^2(\mathbb {D})$$ A 2 ( D ) be the Bergman space that consists of those analytic functions on $$\mathbb {D}$$ D that are of integrable square modulus with respect to the normalized area measure. Let $$\varphi : \mathbb {D} \rightarrow \mathbb {D}$$ φ : D → D be an automorphism of the disk and consider $$C_\varphi f=f \circ \varphi $$ C φ f = f ∘ φ the operator defined from $$\mathcal {A}^2(\mathbb {D})$$ A 2 ( D ) onto itself. Consider the unitary operator $$U_\varphi f = \varphi ^\prime f \circ \varphi $$ U φ f = φ ′ f ∘ φ . Then if $$f \in \mathcal {A}^2(\mathbb {D})$$ f ∈ A 2 ( D ) is even and $$U_\varphi f$$ U φ f is odd, then f is the zero function. The same is true if $$f \in \mathcal {A}^2(\mathbb {D})$$ f ∈ A 2 ( D ) is odd and $$U_\varphi f$$ U φ f is even. Similar results can be proved for the Hardy space of the unit disk, that is, the space of analytic functions on $$\mathbb {D}$$ D , whose Taylor coefficients are of summable square modulus. The result remains true for the Dirichlet space, that is, the space of analytic functions on $$\mathbb {D}$$ D , whose derivatives are in $$\mathcal {A}^2(\mathbb {D})$$ A 2 ( D ) .