On the chaoticity of some tensor product weighted backward shift operators acting on some tensor product Fock-Bargmann spaces

Abstract

In Advances in Mathematical Physics (2011) we showed that the weighted shift $z^{p}\frac{d^{p+1}}{dz^{p+1}}$ $(p=0, 1, 2, ...)$ acting on classical Bargmann space $\mathbb{B}_{p}$ is chaotic operator. In Journal of Mathematical physics (2014), we constructed an chaotic weighted shift $\mathbb{M}^{*^{p}}\mathbb{M}^{p+1}$ $(p=0, 1, 2, ...)$ on some lattice Fock-Bargmann $\mathbb{E}_{p}^{\alpha}$ generated by the orthonormal basis $e_{m}^{(\alpha,p)}(z) = e_{m}^{\alpha} ; m=p, p+1, ...$ where $e_{m}^{\alpha}(z) = (\frac{2\nu}{\pi})^{1/4}e^{\frac{\nu}{2}z^{2}}e^{-\frac{\pi^{2}}{\nu}(m +\alpha)^{2} +2i\pi(m +\alpha)z}; m \in \mathbb{N}$ with $\nu, \alpha$ are real numbers; $\nu > 0$, $\mathbb{M}$ is an weighted shift and $\mathbb{M^{*}}$ is the adjoint of the $\mathbb{M}$. In this paper we study the chaoticity of tensor product $\mathbb{M}^{*^{p}}\mathbb{M}^{p+1}\otimes z^{p}\frac{d^{p}}{dz^{p+1}}$ $(p=0, 1, 2, ...)$ acting on $\mathbb{E}_{p}^{\alpha}\otimes \mathbb{B}_{p}$.