Multivariable Weighted Composition Operators: Lack of Point Spectrum, and Cyclic Vectors

Abstract

We study weighted composition operators T _α,ω on L ²([0, 1]^d) where d ≥ 1, defined by

$$ T_{\alpha ,\omega } f(x_1 , \ldots ,x_d ) = \omega (x_1 , \ldots ,x_d )f(\{ x_1 + \alpha _1 \} , \ldots ,\{ x_d + \alpha _d \} ), $$

where α=(α₁,...α _d )∈ℝ^d and where. denotes the fractional part.

In the case where a is an irrational vector, we give a new and larger class of weights ω for which the point spectrum of T _α,ω is empty. In the case of α∈ℚ^d and ω(x ₁, ..., x _d) = x ₁ ...x _d, we give a complete characterization of the cyclic vectors of T _α,ω.