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

TL;DRAbstract

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.

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