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

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