Diagonally Canonical and Related Gaussian Random Elements

Abstract

We call a Gaussian random element $\eta$ in a Banach space $X$ with a Schauder basis ${\bf e}=(e_n)$ diagonally canonical (for short, $D$-canonical) with respect to ${\bf e}$ if the distribution of $\eta$ coincides with the distribution of a random element having the form $B\xi$, where $\xi$ is a Gaussian random element in $X$, whose ${\bf e}$-components are stochastically independent and $B: X\to X$ is a continuous linear mapping. In this paper we show that if $X=l_p$, $1\le p<\infty$ and $p\neq2,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X,$ which is not $D$-canonical with respect to the natural basis of $X$. We derive this result in the case when $X=l_p$, $2< p<\infty$, or $X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if $X=l_p,$ $2<p<\infty,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X$ such that the distribution of $\eta$ does not coincide with the distribution of the sum of almost surely convergent in $X$ series $\sum_{n=1}^\infty{}x_ng_n$, where $(x_n)$ is an unconditionally summable sequence of elements of $X$ and $(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.