Random quadralinear forms and schur product on tensors

摘要

In this work, we made progress on the problem that l_r directX l_p directX l_q is a Banach algebra under schur product. Our results extend Tonge's results. We also obtained estimates for the norm of the random quadralinear form A: l_r~M x l_p~N x l_q~K x l_s~H →C, defined by: A(e_i, e_j, e_k, e_s)= a_(ijks), where the (a_(ijks))'s are uniformly bounded, independent, mean zero random variables. We proved that under some conditions l_r directX l_p directX l_q directX l_s is not a Banach algebra under schur product.