Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.