Tractability of tensor product linear operators in weighted Hilbert spaces (Q2761604)

The authos studies tractability of linear operators \(S:=S_1 \otimes\cdots \otimes S_d\) where \(S_i\) is a linear operator acting in a Hilbert space \(H_1\oplus \gamma_i^{-1} H_2\), \(0< \gamma_i\leq 1\), \(1\leq i\leq d\), and naturally generated by a single operator \(\widetilde S\) acting in \(H_1\oplus H_2\). Tractability means that a minimal numbers of evaluations needed to reduce the initial error by a factor of \(\varepsilon\) in the (above) \(d\)-dimensional case has a polynomial bound both in \(\varepsilon^{-1}\) and \(d\). The main result, roughly speaking, asserts that \(S\) is tractable only if \(\dim (\widetilde S(H_1)) =1\).