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

TL;DR

This paper investigates the chaotic behavior of tensor product operators combining weighted shift and differential operators on tensor product Fock-Bargmann spaces, extending previous results on chaos in these function spaces.

Contribution

It introduces the analysis of chaos for tensor product operators involving weighted shifts and differential operators on tensor product Fock-Bargmann spaces, a novel extension of prior work.

Findings

Tensor product operators exhibit chaotic behavior under certain conditions.

Extension of chaos results to tensor product of weighted shift and differential operators.

Provides new insights into operator dynamics on complex function 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 $\mathbb{E}_{p}^{\alpha}\otimes \mathbb{B}_{p}$.