Dominated Bilinear Forms and 2-homogeneous Polynomials

Abstract

The main goal of this note is to establish a connection between the cotype of the Banach space $X$ and the parameters $r$ for which every 2-homogeneous polynomial on $X$ is $r$-dominated. Let $\cot X$ be the infimum of the cotypes assumed by $X$ and $(\cot X{)}^{\ast }$ be its conjugate. The main result of this note asserts that if $\cot X>2$, then for every $1\le r<(\cot X{)}^{\ast }$ there exists a non-$r$-dominated 2-homogeneous polynomial on $X$.