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

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Let mathbb {D} denote the unit disk of the complex plane and let mathcal {A}^2(mathbb {D}) be the Bergman space that consists of those analytic functions on mathbb {D} that are of integrable square modulus with respect to the normalized area measure. Let varphi : mathbb {D} rightarrow mathbb {D} be an automorphism of the disk and consider C_varphi f=f circ varphi the operator defined from mathcal {A}^2(mathbb {D}) onto itself. Consider the unitary operator U_varphi f = varphi ^prime f circ varphi . Then if f in mathcal {A}^2(mathbb {D}) is even and U_varphi f is odd, then f is the zero function. The same is true if f in mathcal {A}^2(mathbb {D}) is odd and U_varphi 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}, 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}, whose derivatives are in mathcal {A}^2(mathbb {D}).