On Convergence Properties of Tensor Products of Some Operator Sequences

Abstract

We consider sequences of compact bounded linear operators \(U_n:L^p(0,1)\rightarrow ~L^p(0,1)\) with certain convergence properties. Several divergence theorems for multiple sequences of tensor products of these operators are proved. These theorems in particular imply that \(L\log ^{d-1} L\) is the optimal Orlicz space guaranteeing almost everywhere summability of rectangular partial sums of multiple Fourier series in general orthogonal systems.