Matrix inequalities with applications to the theory of iterated kernels (Q1863589)

For any matrix \(A\) let \(s(A)\) denote the sum of its entries. Define \(A^{(2k)}=(AA^t)^k A^{(2k+1)}=(AA^t)^kA\), where \(A^t\) denotes the transpose of \(A\) and \(k\geq 1\) is any integer. The author proves the following Theorem. Let \(A\) be an \(m\times n\) matrix with nonnegative real entries. Then for every integer \(k\geq 1,\) the following matrix inequalities hold: \[ s(A)^{2k}\leq m^{k-1}n^k s(A^{(2k)})s(A)^{2k+1}\leq m^kn^k s(A^{(2k+1)}). \] The obtained inequalities generalize the matrix inequality \(s(A^3)\leq mn s(AA^tA)\) which was proved by \textit{F. V. Atkinson, G. A. Watterson} and \textit{P. A. P. Moran} [Q. J. Math. Oxf., II. Ser. 11, 137-140 (1960; Zbl 0136.24905)]. The authors give some applications of their results to the theory of iterated kernels.