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 of a certain operator. It is known that if λ1 = 1 and λ2 ∈ (0, 1) then λn = o((lnn)−2), as n → ∞, is a necessary condition for a problem to be weakly tractable. We show this is a sufficient condition as well.