  We study the approximation of compact linear operators defined over certain
weighted tensor product Hilbert spaces. The information complexity is defined
as the minimal number of arbitrary linear functionals which is needed to obtain
an $\varepsilon$-approximation for the $d$-variate problem. It is fully
determined in terms of the weights and univariate singular values. Exponential
tractability means that the information complexity is bounded by a certain
function which depends polynomially on $d$ and logarithmically on
$\varepsilon^{-1}$. The corresponding un-weighted problem was studied recently
by Hickernell, Kritzer and Wo\'{z}niakowski with many negative results for
exponential tractability. The product weights studied in the present paper
change the situation. Depending on the form of polynomial dependence on $d$ and
logarithmic dependence on $\varepsilon^{-1}$, we study exponential strong
polynomial, exponential polynomial, exponential quasi-polynomial, and
exponential $(s,t)$-weak tractability with $\max(s,t)\ge1$. For all these
notions of exponential tractability, we establish necessary and sufficient
conditions on weights and univariate singular values for which it is indeed
possible to achieve the corresponding notion of exponential tractability. The
case of exponential $(s,t)$-weak tractability with $\max(s,t)<1$ is left for
future study.
