The Kagan characterization theorem on Banach spaces

A. Kagan introduced classes of distributions $\mathcal{D}_{m,k}$ in $m$-dimensional space $\mathbb{R}^m$. He proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belong to the class $\mathcal{D}_{m,m-1}$ then the random variables are Gaussian. If $m=2$ then the Kagan theorem implies the well-known Darmois-Skitovich theorem, where the Gaussian distribution is characterized by the independence of two linear forms of $n$ independent random variables. In the paper we describe Banach spaces where the analogue of the Kagan theorem is valid.