On the tractability of linear tensor product problems in the worst case

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues _{λi}i∈N${\left\{{\lambda }_i\right\}}_{i\in \mathbb{N}}$ of a certain operator. It is known that if _λ1=1${\lambda }_1=1$ and _λ2∈(0,1)${\lambda }_2\in (0,1)$ then _λn=o((lnn)⁻²)${\lambda }_n=o\left({(lnn)}^{-2}\right)$, as n→∞$n\to \infty$, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.