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