Abstract
For any real number r, which is greater than or equal to 1, let l_r denote the Banach space of all complex sequences X such that the r-norm of X, denoted ||X||_r, is finite. Given any matrix A with every entry in A is greater than or equal to 0. For any real numbers p and q, p is greater than 1 and q is greater than or equal to 1, let the p, q-norm of A, denoted ||A||_{p, q}, satisfy the following equality: ||A||_{p, q} = sup { ||AX||_q : ||X||_p=1, X belongs to l_p}，where AX is defined by the product of the matrix A and the transpose of the sequence X. In this paper, we replace the problem posed by G. Bennett to a more general problem. The purpose of this paper to find when is the norm of a matrix determined by its action on decreasing sequences. Our results generalize [CLO, Lemma 2.4], and give a partial solution to [B, Problem 7.23]. Our proof is based on the Lagrange multiplier Theorem, which is completely different from the one in [CLO].