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 $z^p\frac{d^{p+1}}{d{z}^{p+1}}$ $(p=0,1,2,...)$ acting on classical Bargmann space ${\mathbb{B}}_p$ is chaotic operator. In Journal of Mathematical physics (2014), we constructed an chaotic weighted shift ${\mathbb{M}}^{{\ast }^p}{\mathbb{M}}^{p+1}$ $(p=0,1,2,...)$ 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 ${\mathbb{M}}^{\ast }$ is the adjoint of the $\mathbb{M}$. In this paper we study the chaoticity of tensor product ${\mathbb{M}}^{{\ast }^p}{\mathbb{M}}^{p+1}\otimes z^p\frac{d^p}{d{z}^{p+1}}$ $(p=0,1,2,...)$ acting on ${\mathbb{E}}_p^{\alpha }\otimes {\mathbb{B}}_p$.