Existence of product vectors and their partial conjugates in a pair of spaces

Young-Hoon Kiem, Seung-Hyeok Kye, Jungseob Lee

Published 2011-07-06, updated 2011-11-22Version 3

Let $D$ and $E$ be subspaces of the tensor product of the $m$ and $n$ dimensional complex spaces, with codimensions $k$ and \ell$, respectively. We show that if $k+\ell<m+n-2$ then there must exist a product vector in $D$ whose partial conjugate lies in $E$. If $k+\ell >m+n-2$ then there may not exist such a product vector. If $k+\ell=m+n-2$ then both cases may occur depending on $k$ and $\ell$.