Cyclic vectors in Fock-type spaces in multi-variable case

Abstract

This manuscript concerns with cyclic vectors in the Fock-type spaces \({L^{p}_{a}}(\mathbb C^d,s,\alpha )\) of multi-variable cases, with positive parameters \(s,\alpha \) and \(p\ge 1\). The one-variable case has been settled by the authors. Here, it is shown that for a positive number \(s\not \in \mathbb {N}\), a function f in the Fock-type space \({L^{p}_{a}}(\mathbb C^d,s,\alpha )\) is cyclic if and only if f is non-vanishing. However, the case of s being a positive integer turns out to be more complicated. Different techniques and methods are developed in multi-variable cases for a complete characterization of cyclic vectors in \({L^{p}_{a}}(\mathbb C^d,s,\alpha )\) for positive integers s.